THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


\yf?  •s^jv-faivArfo+v^-tup 


GEOMETRICAL  RESEARCHES 


ON 


THE  THEORY  OF  PARALLELS 


BY 

NICHOLAS  LOBACHEVSKI 

Imperial  Russian  Real  Councillor  of  State  and  Regular  Professor 
of  Mathematics  in  the  University  of  Rasan 

BERLIN,  1840 


Translated  from  the  Original 
BY 

GEORGE  BRUCE  HALSTED 

A.  M.,  Ph.  D.,  Ex-Fellow  of  Princeton  and 
Johns-Hopkins  University 


NEW  EDITION 


LA  SALLE,  ILLINOIS 

OPEN  COURT  PUBLISHING  COMPANY 

1914 


Plonographed  by  John  S.  Swift  Co.,  Inc.,  Chicago.  St.  Louis,  New  York,  Cincinnati 


Copyright  by 

The  Open  Court  Publishing  Co., 
Chicago,    1914. 


Mathematical 
Sciences 
Library 


'Q  PR  1?  PA  PI? 
o  rKMAUti, 


Lobachevski  was  the  first  man  ever  to  publish  a  non-Euclidean  geom- 
etry. 

Of  the  immortal  essay  now  first  appearing  in  English  Gauss  said,  "The 
author  has  treated  the  matter  with  a  master-hand  and  in  the  true  geom- 
eter's spirit.  I  think  I  ought  to  call  your  attention  to  this  book,  whose 
perusal  can  not  fail  to  give  you  the  most  vivid  pleasure." 

Clifford  says,  "It  is  quite  simple,  merely  Euclid  without  the  vicious 
assumption,  but  the  way  things  come  out  of  one  another  is  quite  lovely." 
*  *  *  "What  Vesalius  was  to  Galen,  what  Copernicus  was  to  Ptolemy, 
that  was  Lobachevski  to  Euclid." 

Says  Sylvester,  "In  Quaternions  the  example  has  been  given  of  Al- 
gebra released  from  the  yoke  of  the  commutative  principle  of  multipli- 
cation— an  emancipation  somewhat  akin  to  Lobachevski's  of  Geometry 
from  Euclid's  noted  empirical  axiom." 

Cayley  says,  "It  is  well  known  that  Euclid's  twelfth  axiom,  even  in 
Playf air's  form  of  it,  has  been  considered  as  needing  demonstration; 
and  that  Lobachevski  constructed  a  perfectly  consistent  theory,  where- 
in this  axiom  was  assumed  not  to  hold  good,  or  say  a  system  of  non- 
Euclidean  plane  geometry.  There  is  a  like  system  of  non-Euclidean  solid 
geometry." 

GEORGE  BRUCE  HALSTED. 
2407  San  Marcos  Street, 

Austin,  Texas. 
May  1,  1891. 


TRANSLATOR'S  INTRODUCTION. 


"Prove  all  things,  hold  fast  that  which  is  good,"  does  not  mean  dem- 
onstrate everything.  "Prom  nothing  assumed,  nothing  can  be  proved. 
"Geometry  without  axioms,"  was  a  book  .which  went  through  several 
editions,  and  still  has  historical  value.  But  now  a  volume  with  such  a 
title  would,  without  opening  it,  be  set  down  as  simply  the  work  of  a 
paradoxer. 

The  set  of  axioms  far  the  most  influential  in  the  intellectual  history 
of  the  world  was  put  together  in  Egypt;  but  really  it  owed  nothing  to 
the  Egyptian  race,  drew  nothing  from  the  boasted  lore  of  Egypt's 
priests. 

The  Papyrus  of  the  Rhind,  belonging  to  the  British  Museum,  -but 
given  to  the  world  by  the  erudition  of  a  German  Egyptologist,  Eisen- 
lohr,  and  a  German  historian  of  mathematics,  Cantor,  gives  us  more 
knowledge  of  the  state  of  mathematics  in  ancient  Egypt  than  all  else 
previously  accessible  to  the  modern  world.  Its  whole  testimony  con- 
firms with  overwhelming  force  the  position  that  Geometry  as  a  science, 
strict  and  self-conscious  deductive  reasoning,  was  created  by  the  subtle 
intellect  of  the  same  race  whose  bloom  in  art  still  overawes  us  in  the 
Venus  of  Milo,  the  Apollo  Belvidere,  the  Laocoon. 

In  a  geometry  occur  the  most  noted  set  of  axioms,  the  geometry  of 
Euclid,  a  pure  Greek,  professor  at  the  University  of  Alexandria. 

Not  only  at  its  very  birth  did  this  typical  product  of  the  Greek  genius 
assume  sway  as  ruler  in  the  pure  sciences,  not  only  does  its  first  efflor- 
escence carry  us  through  the  splendid  days  of  Theon  and  Hypatia,  but 
unlike  the  latter,  fanatics  can  not  murder  it;  that  dismal  flood,  the  dark 
ages,  can  not  drown  it.  Like  the  phoeniz  of  its  native  Egypt,  it  rises 
with  the  new  birth  of  culture.  An  Anglo-Saxon,  Adelard  of  Bath, 
finds  it  clothed  in  Arabic  vestments  in  the  land  of  the  Alhambra.  Then 
clothed  in  Latin,  it  and  the  new-born  printing  press  confer  honor  on 
each  other.  Finally  back  again  in  its  original  Greek,  it  is  published 
first  in  queenly  Basel,  then  in  stately  Oxford.  The  latest  edition  in 
Greek  is  from  Leipsic's  learned  presses. 


6  THEORY    OP    PARALLELS. 

How  the  first  translation  into  our  cut-and-thrust,  survival-of-the-fittest 
English  was  made  from  the  Greek  and  Latin  by  Henricus  Billingsly, 
Lord  Mayor  of  London,  and  published  with  a  preface  by  John  Dee  the 
Magician,  may  be  studied  In  the  Library  of  our  own  Princeton,  where 
they  have,  by  some  strange  chance,  Billingsly's  own  copy  of  the  Arabic- 
Latin  version  of  Campanus  bound  with  the  Editio  Princeps  in  Greek 
and  enriched  with  his  autograph  emendations.  Even  to-day  in  the  vast 
system  of  examinations  set  by  Cambridge,  Oxford,  and  the  British  gov- 
ernment, no  proof  will  be  accepted  which  infringes  Euclid's  order,  a 
sequence  founded  upon  his  set  of  axioms. 

The  American  ideal  is  success.  In  twenty  years  the  American  maker 
expects  to  be  improved  upon,  superseded.  The  Greek  ideal  was  per- 
fection. The  Greek  Epic  and  Lyric  poets,  the  Greek  sculptors,  remain 
unmatched.  The  axioms  of  the  Greek  geometer  remained  unquestioned 
for  twenty  centuries. 

How  and  where  doubt  came  to  look  toward  them  is  of  no  ordinary 
interest,  for  this  doubt  was  epoch-making  in  the  history  of  mind. 

Among  Euclid's  axioms  was  one  differing  from  the  others  in  pro- 
lixity, whose  place  fluctuates  in  the  manuscripts,  and  which  is  not  used 
in  Euclid's  first  twenty-seven  propositions.  Moreover  it  is  only  then 
brought  in  to  prove  the  inverse  of  one  of  these  already  demonstrated. 

All  this  suggested,  at  Europe's  renaissance,  not  a  doubt  of  the  axiom, 
but  the  possibility  of  getting  along  without  it,  of  deducing  it  from  the 
other  axioms  and  the  twenty -seven  propositions  already  proved.  Euclid 
demonstrates  things  more  axiomatic  by  far.  He  proves  what  every  dog 
knows,  that  any  two  sides  of  a  triangle  are  together  greater  than  the 
third.  Yet  when  he  has  perfectly  proved  that  lines  making  with  a 
transversal  equal  alternate  angles  are  parallel,  in  order  to  prove  the  in- 
verse,  that  parallels  cut  by  a  transversal  make  equal  alternate  angles,  he 
brings  in  the  un wieldly  postulate  or  axiom: 

"  If  a  straight  line  meet  two  straight  lines,  so  as  to  make  the  two  in- 
terior angles  on  the  same  side  of  it  taken  together  less  than  two  right 
angles,  these  straight  lines,  being  continually  produced,  shall  at  length 
meet  on  that  side  on  which  are  the  angles  which  are  less  than  two  right 
angles." 

Do  you  wonder  that  succeeding  geometers  wished  by  demonstration 
to  push  this  un  wieldly  thing  from  the  set  of  fundamental  axioms. 


TRANSLATOR  8    INTRODUCTION.  7 

Numerous  and  desperate  were  the  attempts  to  deduce  it  from  reason- 
ings about  the  nature  of  the  straight  line  and  plane  angle.  In  the 
"  Encyclopcedie  der  Wissenschaften  und  Kunste;  Von  Erech  und  Gru- 
ber;"  Leipzig,  1838;  under  "Parallel,"  Sohncke  says  that  in  mathe- 
matics there  is  nothing  over  which  so  much  has  been  spoken,  written, 
and  striven,  as  over  the  theory  of  parallels,  and  all,  so  far  (up  to  his 
time),  without  reaching  a  definite  result  and  decision. 

Some  acknowledged  defeat  by  taking  a  new  definition  of  parallels,  as 
for  example  the  stupid  one,  "Parallel  lines  are  everywhere  equally  dis- 
tant," still  given  on  page  33  of  Schuyler's  Geometry,  which  that  author, 
like  many  of  his  unfortunate  prototypes,  then  attempts  to  identify  with 
Euclid's  definition  by  pseudo-reasoning  which  tacitly  assumes  Euclid's 
postulate,  e.  g.  he  says  p.  35:  "For,  if  not  parallel,  they  are  not  every- 
where equally  distant;  and  since  they  lie  in  the  same  plane;  must  ap- 
proach when  produced  one  way  or  the  other;  and  since  straight  lines 
continue  in  the  same  direction,  must  continue  to  approach  if  produced 
farther,  and  if  sufficiently  produced,  must  meet."  This  is  nothing  but 
Euclid's  assumption,  diseased  and  contaminated  by  the  introduction  of 
the  indefinite  term  "direction." 

How  much  better  to  have  followed  the  third  class  of  his  predecessors 
who  honestly  assume  a  new  axiom  differing  from  Euclid's  in  form  if 
not  in  essence.  Of  these  the  best  is  that  catted  Playfair's;  "Two  lines 
which  intersect  can  not  both  be  parallel  to  the  same  line." 

The  German  article  mentioned  is  followed  by  a  carefully  prepared 
list  of  ninety-two  authors  on  the  subject.  In  English  an  account  of 
like  attempts  was  given  by  Perronet  Thompson,  Cambridge,  1833,  and 
is  brought  up  to  date  in  the  charming  volume,  "Euclid  and  his  Modern 
Rivals,"  by  C.  L.  Dodgson,  late  Mathematical  Lecturer  of  Christ  Church, 
Oxford,  the  Lewis  Carroll,  author  of  Alice  in  Wonderland. 

All  this  shows  how  ready  the  world  was  for  the  extraordinary  flaming- 
forth  of  genius  from  different  parts  of  the  world  which  was  at  once  to 
overturn,  explain,  and  remake  not  only  all  this  subject  but  as  conse- 
quence all  philosophy,  all  ken-lore.  As  was  the  case  with  the  dis- 
covery of  the  Conservation  of  Energy,  the  independent  irruptions 
of  genius,  whether  in  Russia,  Hungary,  Germany,  or  even  in  Canada 
gave  everywhere  the  same  results. 

At  first  these  results  were  not  fully  understood  even  by  the  brightest 


8  THEORY    OF    PARALLELS. 

intellects.  Thirty  years  after  the  publication  of  the  'book  he  mentions, 
we  see  the  brilliant  Clifford  writing  from  Trinity  College,  Cambridge, 
April  2,  1870,  "Several  new  ideas  have  come  to  me  lately:  First  I 
have  procured  Lobachevski,  'Etudes  Geom6triques  sur  la  Theorie 
des  Parallels'  -  -  -  a  small  tract  of  which  Gauss,  therein  quoted, 
says :  L'auteur  a  traits  la  matiere  en  main  de  maltre  et  avec  le  veritable 
esprit  geometrique.  Je  crois  devoir  appeler  votre  attention  sur  ce  livre, 
dont  la  lecture  ne  peut  manquer  de  vous  causer  le  plus  vif  plaisir.'" 
Then  says  Clifford:  "It  is  quite  simple,  merely  Euclid  without  the 
vicious  assumption,  but  the  way  the  things-  come  out  of  one  another  is 
quite  lovely." 

The  first  axiom  doubted  is  called  a  "vicious  assumption,"  soon  no 
man  sees  more  clearly  than  Clifford  that  all  are  assumptions  and  none 
vicious.  He  had  been  reading  the  French  translation  by  Hoiiel,  pub- 
lished in  1866,  of  a  little  book  of  61  pages  published  in  1840  in  Berlin 
under  the  title  Geometrische  Untersuchungen  zur  Theorie  der  Parallel- 
linien  by  Nicolas  Lobachevski  (1793-1856),  the  first  public  expression 
of  whose  discoveries,  however,  dates  back  to  a  discourse  at  Kasan  on 
February  12,  1826. 

Under  this  commonplace  title  who  would  have  suspected  the  dis- 
covery of  a  new  space  in  which  to  hold  our  universe  and  ourselves. 

A  new  kind  of  universal  space;  the  idea  is  a  hard  one.  To  name  it, 
all  the  space  in  which  we  think  the  world  and  stars  live  and  move  and 
have  their  being  was  ceded  to  Euclid  as  his  by  right  of  pre-emption, 
description,  and  occupancy;  then  the  new  space  and  its  quick-following 
fellows  could  be  called  Non-Euclidean. 

Gauss  in  a  letter  to  Schumacher,  dated  Nov.  28,  1846,  mentions  that 
as  far  back  as  1792  he  had  started  on  this  path  to  a  new  universe. 
Again  he  says:  "La  geometric  non-euclidienne  ne  renferme  en  elle 
rien  de  contradictoire,  quoique,  8.  premiere  vue,  beaucoup  de  ses  rfeul- 
tats  aien  1'air  de  paradoxes.  Ces  contradictions  apparents  doivent  etre 
regardees  comme  Peffet  d'une  illusion,  due  a  1'habitude  que  nous  avons 
prise  de  bonne  heure  de  considerer  la  geometric  euclidienne  comme 
rigoureuse." 

But  here  we  see  in  the  last  word  the  same  imperfection  of  view  as  in 
Clifford's  letter.  The  perception  has  not  yet  come  that  though  the  non- 
Euclidean  geometry  is  rigorous,  Euclid  is  not  one  whit  less  so. 


TRANSLATOR'S    INTRODUCTION.  V 

A  former  friend  of  Gauss  at  Goettingen  was  the  Hungarian  Wolfgang 
Bolyai.  His  principal  work,  published  by  subscription,  has  the  follow- 
ing title: 

Tentamen  Juventutem  studiosam  in  elementa  Matheseos  purae,  ele- 
meutaris  ac  sublimiorls,  methodo  intuitiva,  evidentiaque  huic  propria,  in- 
troducendi.  Tomus  Primus,  1832;  Secundus,  1833.  8vo.  Maros-Va- 
sarhelyini. 

In  the  first  volume  with  special  numbering,  appeared  the  celebrated 
Appendix  of  his  son  John  Bolyai  with  the  following  title : 

APPENDIX. 

SCIENTIAM  SPATII  absolute  veram  exhibens:  a  veritate  aut  falsitate 
Axiomatis  XI  Euclidei  (a  priori  haud  unquam  decidenda)  independen- 
tem.  Auctore  JOHANNE  BOLYAI  de  eadem,  Geometrarum  in  Exercitu 
Caesareo  Regio  Austriaco  Castrensium  Capitaneo.  (26  pages  of  text). 

This  marvellous  Appendix  has  been  translated  into  French,  Italian, 
English  and  German. 

In  the  title  of  Wolfgang  Bolyai's  last  work,  the  only  one  he  com- 
posed in  German  (88  pages  of  text,  1851),  occurs  the  following: 

"und  da  die  Frage,  06  zwey  von  der  dritten  geschnittene  Geraden. 
wenn  die  summe  der  inneren  Wtnkel  nicht=2R,  sich  schneiden  oder 
nichtf  niemand  auf  der  Erde  ohne  ein  Axiom  (wie  Euclid  das  XI) 
aufzustellen,  beantworten  wird;  die  davon  unabhaengige  Geometric 
abzusondern ;  und  eine  auf  die  Ja-Antwort,  andere  auf  das  Nein  so  zu 
bauen,  dass  die  Formeln  der  letzen,  auf  ein  Wink  auch  in  der  ersten 
gultig  seyen." 

The  author  mentions  Lobachevski's  Geometrische  Untersuchungen, 
Berlin,  1840,  and  compares  it  with  the  work  of  his  son  John  Bolyai, 
"au  sujet  duquel  il  dit :  'Quelques  exemplaires  de  1'ouvrage  publiS  ici 
ont  £t6  envoyfe  a  cette  6poque  S  Vienne,  a  Berlin,  a  Gcettingue.  .  .  De 
<l!oettingue  le  ggant  math^matique,  [Gauss]  qui  du  sommet  des  hauteurs 
embrasse  du  m6me  regard  les  astres  et  la  profondeur  des  ablmes,  a  £crit 
qu'il  ^tait  ravi  de  voir  execute  le  travail  qu'fl  avait  commence  pour  le 
laisser  apres  lui  dans  ses  papiers.' " 

In  fact  this  first  of  the  Non-Euclidean  geometries  accepts  all  of  Eu- 
clid's axioms  but  the  last,  which  it  flatly  denies  and  replaces  by  its  con- 
tradictory, that  the  sum  of  the  interior  angles  made  on  the  same  side  of 


10  THEORY    OF    PARALLELS. 

a  transversal  by  two  straight  lines  may  be  leas  than  a  straight  angle 
without  the  lines  meeting.  A  perfectly  consistent  and  elegant  geometry 
then  follows,  in  which  the  sum  of  the  angles  of  a  triangle  is  always  less 
than  a  straight  angle,  and  not  every  triangle  has  its  rertices  coacytlit. 


THEORY  OF  PARALLELS. 


In  geometry  I  find  certain  imperfections  which  I  hold  to  be  the  rea- 
son why  this  science,  apart  from  transition  into  analytics,  can  as  yet 
make  no  advance  from  that  state  in  which  it  has  come  to  us  from  Euclid. 

As  belonging  to  these  imperfections,  I  consider  the  obscurity  in  the 
fundamental  concepts  of  the  geometrical  magnitudes  and  in  the  manner 
and  method  of  representing  the  measuring  of  these  magnitudes,  and 
finally  the  momentous  gap  in  the  theory  of  parallels,  to  fill  which  all  ef- 
forts of  mathematicians  have  been  so  far  in  vain. 

For  this  theory  Legendre's  endeavors  have  done  nothing,  since  he 
was  forced  to  leave  the  only  rigid  way  to  turn  into  a  side  path  and  take 
refuge  in  auxiliary  theorems  which  he  illogically  strove  to  exhibit  as 
necessary  axioms.  My  first  essay  on  the  foundations  of  geometry  I  pub- 
lished in  the  Kasan  Messenger  for  the  year  1829.  In  the  hope  of  having 
satisfied  all  requirements,  I  undertook  hereupon  a  treatment  of  the  whole 
of  this  science,  and  published  my  work  in  separate  parts  in  the  "  Ge- 
lehrten  Schriflen  der  Universilaet  Kasan"  for  the  years  1836,  1837,  1838, 
under  the  title  "New  Elements  of  Geometry,  with  a  complete  Theory 
of  Parallels."  The  extent  of  this  work  perhaps  hindered  my  country- 
men from  following  such  a  subject,  which  since  Legendre  had  lost  its 
interest.  Yet  I  am  of  the  opinion  that  the  Theory  of  Parallels  should 
not  lose  its  claim  to  the  attention  of  geometers,  and  therefore  I  aim  to 
give  here  the  substance  of  my  investigations,  remarking  beforehand  that 
contrary  to  the  opinion  of  Legendre,  all  other  imperfections — for  ex- 
ample, the  definition  of  a  straight  line — show  themselves  foreign  here 
and  without  any  real  influence  on  the  theory  of  parallels. 

In  order  not  to  fatigue  my  reader  with  the  multitude  of  those  theo- 
rems whose  proofs  present  no  difficulties,  I  prefix  here  only  those  of 
which  a  knowledge  is  necessary  for  what  follows. 

1.  A  straight  line  fits  upon  itself  in  all  its  positions.  By  this  I  mean 
that  during  the  revolution  of  the  surface  containing  it  the  straight  line 
does  not  change  its  place  if  it  goes  through  two  unmoving  points  in  the 
surface:  (i.  e.,  if  we  turn  the  surface  containing  it  about  two  points  of 
the  line,  the  line  does  not  move.) 


12  THEORY    OF    PARALLELS. 

2.  Two  straight  lines  can  not  intersect  in  two  points. 

3.  A  straight  line  sufficiently  produced  both  ways  must  go  out 
beyond  all  bounds,  and  in  such  way  cuts  a  bounded  plain  into  two  parts. 

4.  Two  straight  lines  perpendicular  to  a  third  never  intersect,  how 
far  soever  they  be  produced. 

6.  A  straight  line  always  cuts  another  in  going  from  one  side  of  it 
over  to  the  other  side:  («'.  e.,  one  straight  line  must  cut  another  if  it 
has  points  on  both  sides  of  it.) 

6.  Vertical  angles,  where  the  sides  of  one  are  productions  of  the 
sides  of  the  other,  are  equal.     This  holds  of  plane  rectilineal  angles 
among  themselves,  as  also  of  plane  surface  angles :    (i.  e.,  dihedral  angles.) 

7.  Two  straight  lines  can  not  intersect,  if  a  third  cuts  them  at  the 
same  angle. 

8.  In  a  rectilineal  triangle  equal  sides  lie  opposite  equal  angles,  and 
inversely. 

9.  In  a  rectilineal  triangle,  a  greater  side  lies  opposite  a  greater 
angle.     In  a  right-angled  triangle  the  hypothenuse  is  greater  than  either 
of  the  other  sides,  and  the  two  angles  adjacent  to  it  are  acute. 

10.  Rectilineal  triangles  are  congruent  if  they  have  a  side  and  two 
angles  equal,  or  two  sides  and  the  included  angle  equal,  or  two  sides  and 
the  angle  opposite  the  greater  equal,  or  three  sides  equal. 

11.  A  straight   line  which  stands  at  right  angles  upon  two  other 
straight  lines  not  in  one  plane  with  it  is  perpendicular  to  all  straight 
lines  drawn  through  the  common  intersection  point  in  the  plane  of  those 
two. 

12.  The  intersection  of  a  sphere  with  a  plane  is  a  circle. 

13.  A  straight  line  at  right  angles  to  the  intersection  of  two  per- 
pendicular  planes,  and  in  one,  is  perpendicular  to  the  other. 

14. .  In  a  spherical  triangle  equal  sides  lie  opposite  equal  angles,  and 
inversely. 

16.  Spherical  triangles  are  congruent  (or  symmetrical)  if  they  have 
two  sides  and  the  included  angle  equal,  or  a  side  and  the  adjacent  angles 
equal. 

From  here  follow  the  other  theorems  with  their  explanations  and 
proofs. 


• 


THEORY    OF    PARALLELS.  13 

16.  All  straight  lines  which  in  a  plane  go  out  from  a  point  can, 
with  reference  to  a  given  straight  line  in  the  same  plane,  be  divided 
into  two  classes — into  cutting  and  not-cutting. 

The  boundary  lines  of  the  one  and  the  other  class  of  those  lines  will 
be  called  parallel  to  the  given  line.  0 

From  the  point  A  (Fig.  1)  let  fall  upon  the 
line  BC  the  perpendicular  AD,  to  which  again 
draw  the  perpendicular  AE. 

In  the  right  angle  EAD  either  will  all  straight 
lines  which  go  out  from  the  point  A  meet  the 
line  DC,  as  for  example  AF,  or  some  of  them, 
like  the  perpendicular  AE,  will  not  meet  the 
line  DC.  In  the  uncertainty  whether  the  per- 
pendicular AE  is  the  only  line  which  does  not 
meet  DC,  we  will  assume  it  may  be  possible  that  «'' 

there  are  still   other  lines,  for  example  AG,  FIG.  1. 

which  do  not  cut  DC,  how  far  soever  they  may  be  prolonged.  In  pass- 
ing over  from  the  cutting  lines,  as  AF,  to  the  not-cutting  lines,  as  AG, 
we  must  come  upon  a  line  AH,  parallel  to  DC,  a  boundary  line,  upon 
one  side  of  which  all  lines  AG  are  such  as  do  not  meet  the  line  DC, 
while  upon  the  other  side  every  straight  line  AF  cuts  the  line  DC. 

The  angle  HAD  between  the  parallel  HA  and  the  perpendicular  AD 
is  called  the  parallel  angle  (angle  of  parallelism),  which  we  will  here 
designate  by  f]  (p)  for  AD  =  p. 

If  77  (p)  is  a  right  angle,  so  will  the  prolongation  AE'  of  the  perpen- 
dicular AE  likewise  be  parallel  to  the  prolongation  DB  of  the  line  DC, 
in  addition  to  which  we  remark  that  in  regard  to  the  four  right  angles, 
which  are  made  at  the  point  A  by  the  perpendiculars  AE  and  AD, 
and  their  prolongations  AE'  and  AD',  every  straight  line  which  goes 
out  from  the  point  A,  either  itself  or  at  least  its  prolongation,  lies  in  one 
of  the  two  right  angles  which  are  turned  toward  BC,  so  that  except  the 
parallel  EE'  all  others,  if  they  are  sufficiently  produced  both  ways,  must 
intersect  the  line  BC. 

If  77  (p)  <  \  x,  then  upon  the  other  side  of  AD,  making  the  same 
angle  DAK  =  II  (p)  will  lie  also  a  line  AK,  parallel  to  the  prolonga- 
tion DB  of  the  line  DC,  so  that  under  this  assumption  we  must  also 
make  a  distinction  of  sides  in  parallelism. 


14  THEORY    OF    PARALLELS. 

All  remaining  lines  or  their  prolongations  within  the  two  right  angles 
turned  toward  BC  pertain  to  those  that  intersect,  if  they  lie  within  the 
angle  HAK  =  2  H  (p)  between  the  parallels;  they  pertain  on  the  other 
hand  to  the  non-intersecting  AG,  if  they  lie  upon  the  other  sides  of  the 
parallels  AH  and  AK,  in  the  opening  of  the  two  angles  EAH  =  \  jt 
—  /7  (p),  E'AK  =  |  JT  —  77  (p),  between  the  parallels  and  EE'  the  per- 
pendicular to  AD.  Upon  the  other  side  of  the  perpendicular  EE'  will 
in  like  manner  the  prolongations  AH'  and  AK'  of  the  parallels  AH  and 
AK  likewise  be  parallel  to  BC;  the  remaining  lines  pertain,  if  in  the 
angle  K'AH',  to  the  intersecting,  but  if  in  the  angles  K'AE,  H'AE' 
to  the  non-intersecting. 

In  accordance  with  this,  for  the  assumption  77  (p)  =  ^  rc.  the  lines  can 
be  only  intersecting  or  parallel;  but  if  we  assume  that  77 (p)  <  £  TT,  then 
we  must  allow  two  parallels,  one  on  the  one  and  one  on  the  other  side; 
in  addition  we  must  distinguish  the  remaining  lines  into  non-intersect- 
ing and  intersecting. 

For  both  assumptions  it  serves  as  the  mark  of  parallelism  that  the 
line  becomes  intersecting  for  the  smallest  deviation  toward  the  side 
where  lies  the  parallel,  so  that  if  AH  is  parallel  to  DC,  every  line  AP 
cuts  DC,  how  small  soever  the  angle  HAF  may  be. 


THEORY    OP    PARALLELS. 


16 


17.     A  straight  line  maintains  the  characteristic  of  parallelism  at  aU  its 
points. 
Given  AB  (Fig.  2)  parallel  to  CD,  to  which  latter  AC  is  perpendic 


ular.  "We  -will  consider  two  points  taken  at  random  on  the  line  AB  and 
its  production  beyond  the  perpendicular. 

Let  the  point  E  lie  on  that  side  of  the  perpendicular  on  which  AB  is 
looked  upon  as  parallel  to  CD. 

Let  fall  from  the  point  E  a  perpendicular  EK  on  CD  and  so  draw  EP 
that  it  falls  within  the  angle  BEK. 

Connect  the  points  A  and  F  by  a  straight  line,  whose  production  then 
(by  Theorem  1 6)  must  cut  CD  somewhere  in  G.  Thus  we  get  a  triangle 
ACG,  into  which  the  line  EF  goes;  now  since  this  latter,  from  the  con- 
struction,  can  not  cut  AC,  and  can  not  cut  AG  or  EK  a  second  time 
(Theorem  2),  therefore  it  must  meet  CD  somewhere  at  H  (Theorem  3). 

Now  let  E'  be  a  point  on  the  production  of  AB  and  E'K'  perpendic- 
ular to  the  production  of  the  line  CD;  draw  the  line  E'F'  making  so 
small  an  angle  AE'F'  that  it  cuts  AC  somewhere  in  F';  making  the 
same  angle  with  AB,  draw  also  from  A  the  line  AF,  whose  production 
will  cut  CD  in  G  (Theorem  16.) 

Thus  we  get  a  triangle  AGO,  into  which  goes  the  production  of  the 
line  E'F';  since  now  this  line  can  not  cut  AC  a  second  time,  and  also 
can  not  cut  AG,  since  the  angle  BAG  =  BE'G',  (Theorem  7),  therefore 
must  it  meet  CD  somewhere  in  G'. 

Therefore  from  whatever  Doints  E  and  E'  the  lines  EF  and  E'F'  go 
out,  and  however  little  they  may  diverge  from  the  line  AB,  yet  will 
they  always  cut  CD,  to  which  AB  is  parallel 


16 


THEORY    OF    PARALLELS. 


18.  Two  lines  are  always  muttmlly  parallel. 

Let   AC   be  a  perpendicular    on   CD,   to   which   AB   is   parallel 
if  we  draw  from  C  the  line  A 
CE  making  any  acute   angle 
BCD  with  CD,  and  let  fall 
from  A  the  perpendicular  AF 
upon  CE,  we  obtain  a  right- 
angled  triangle  ACF,  in  which 
AC,  being  the  hypothenuse, 
is  greater  than  the  side  AF 
(Theorem  9.) 

Make  AG  =  AF,  and  slide  Fl°-  3- 

the  figure  EFAB  until  AF  coincides  with  AG,  when  AB  and  FE  wil 
take  the  position  AK  and  GH,  such  that  the  angle  BAK  =  FAC,  con 
sequently  AK  must  cut  the  line  DC  somewhere  in  K  (Theorem  1 6),  thu» 
forming  a  triangle  AKC,  on  one  side  of  which  the  perpendicular  GH 
intersects  the  line  AK  in  L  (Theorem  3),  and  thus  determines  the  dis 
tance  AL  of  the  intersection  point  of  the  lines  AB  and  CE  on  the  lin« 
AB  from  the  point  A. 

Hence  it  follows  that  CE  will  always  intersect  AB,  how  small  soever 
may  be  the  angle  BCD,  consequently  CD  is  parallel  to  AB  (Theorem  16.) 

19.  In  a  rectilineal  triangle  the  sum  of  the  three  angles  can  not  be  greater 
than  two  right  angles. 

Suppose  in  the  triangle  ABC  (Fig.  4)  the  sum  of  the  three  angles  k 
equal  to  TT  -f-  a-  then  choose  in  case 
of  the  inequality  of  the  sides  the 
smallest  BC,  halve  it  in  D,  draw 
from  A  through  D  the  line  AD 
and  make  the  prolongation  of  it, 
DE,  equal  to  AD,  then  join  the  A 
point  E   to  the  point   C  by  the 


FIQ.  4. 


straight  line  EC.  In  the  congruent  triangles  ADB  and  CDE,  the  angle 
ABD  =  DCE,  and  BAD  =  DEC  (Theorems  6  and  10);  whence  follows 
that  also  in  the  triangle  ACE  the  sum  of  the  three  angles  must  be  equal 
to  TZ  -f-  a;  but  also  the  smallest  angle  BAG  (Theorem  9)  of  the  triangle 
ABC  in  passing  over  into  the  new  triangle  ACE  has  been  cut  up  into 
the  two  parts  EAC  and  AEC.  Continuing  this  process,  continually 


THEORY    OF    PARALLELS. 


17 


halving  the  side  opposite  the  smallest  angle,  we  must  finally  attain  to  a 
triangle  in  which  the  sum  of  the  three  angles  is  jr  -f-  a,  but  wherein  are 
two  angles,  each  of  which  in  absolute  magnitude  is  less  than  -Ja;  since 
now,  however,  the  third  angle  can  not  be  greater  than  K,  so  must  a  be 
either  null  or  negative. 

20.  If  in  any  rectilineal  triangle  the  sum  of  the  three  angles  is  equal  to 
two  right  angles,  so  is  this  also  the  case  for  every  other  triangle. 

If  in  the  rectilineal  triangle  ABC  (Fig.  5)  the  sum  of  the  three  angles 
=r  ;r,  then  must  at  least  two  of  its  angles,  A  B 

and  C,  be  acute.  Let  fall  from  the  vertex  of 
the  third  angle  B  upon  the  opposite  side  AC 
the  perpendicular  p.  This  will  cut  the  tri- 
angle  into  two  right-angled  triangles,  in  each 
of  which  the  sum  of  the  three  angles  must  also  be  TT,  since  it  can  not  in 
either  be  greater  than  77,  and  in  their  combination  not  less  than  TT. 

So  we  obtain  a  right-angled  triangle  with  the  perpendicular  sides  p 
and  q,  and  from  this  a  quadrilateral  whose  opposite  sides  are  equal  and 
whose  adjacent  sides  p  and  q  are  at  right  angles  (Fig.  6.) 

By  repetition  of  this  quadrilateral  we  can  make  another  with  sides 
np  and  q,  and  finally  a  quadrilateral  ABCD  with  sides  at  right  angles 
to  each  other,  such  that  AB  =  np,  AD  =  mq,  DC  =  np,  BC  =  mq,  where 


FIG.  6. 

m  and  n  are  any  whole  numbers.  Such  a  quadrilateral  is  divided  by 
the  diagonal  DB  into  two  congruent  right-angled  triangles,  BAD  and 
BCD,  in  each  of  which  the  sum  of  the  three  angles  =  jr. 

The  numbers  n  and  m  can  be  taken  sufficiently  great  for  the  right- 
angled  triangle  ABC  (Fig.  7)  whose  perpendicular  sides  AB  =  np,  BC 
=  mq,  to  enclose  within  itself  another  given  (right-angled)  triangle  BDE 

as  soon  as  the  right-angles  fit  each  other. 
2  — par. 


18 


THEORY    OF    PARALLELS. 


Drawing  the  line  DC,  we  obtain  right-angled  triangles  of  which  every 
successive  two  have  a  side  in  common. 

The  triangle  ABC  is  formed  by  the  union  of  the  two  triangles  ACD 
and  DCB,  in  neither  of  which  can  the  earn  of  the  angles  be  greater  than 
TT  ;  consequently  it  must  be  equal  to  x,  in  order  that  the  sum  in  the 
compound  triangle  may  be  equal  to  x. 


Fio.  7. 

In  the  same  way  the  triangle  BDC  consists  of  the  two  triangles  DEC 
and  DBE,  consequently  must  in  DBE  the  sum  of  the  three  angles  be 
equal  to  x,  and  in  general  this  must  be  true  for  every  triangle,  since 
each  can  be  <mt  into  two  right-angled  triangles. 

From  this  it  follows  that  only  two  hypotheses  are  allowable:  Either 
is  the  sum  of  the  three  angles  in  all  rectilineal  triangles  equal  to  x,  or 
this  sum  is  in  all  less  than  x. 

21.  From  a  given  point  we  can  always  draw  a  straight  line  that  shall 
make  with  a  given  straight  line  an  angle  as  small  as  we  choose. 

Let  fall  from  the  given  point  A  (Fig.  8)  upon  the  given  line  BC  the 
A 


E  C 

Fio  8. 

perpendicular  AB;  take  upon  BC  at  random  the  point  D:  draw  the  line 
AD;  make  DE  =  AD,  and  draw  AE. 


THEORY    OF    PARALLELS.  19 

In  the  right-angled  triangle  ABB  let  the  angle  ADB  =  a;  then  must 
in  the  isosceles  triangle  ADE  the  angle  AED  be  either  %a  or  less  (Theo- 
rems 8  and  20).  Continuing  thus  we  finally  attain  to  such  an  angle, 
AEB,  as  is  less  than  any  given  angle. 

22.  If  two  perpendiculars  to  the  same  straight  line  are  parallel  to  each 
other,  then  the  sum  of  the  three  angles  in  ^rectilineal  triangle  is  equal  to  two 
right  angles. 

Let  the  lines  AB  and  CD  (Fig.  9)  be  parallel  to  each  other  and  per- 
pendicular to  AC. 

Draw  from  A  the  lines  AE 
and  AF  to  the  points  E  and  F, 
which  are  taken  on  the  line  CD 
at  any  distances  FC  >  EC  from 
the  point  C. 

Suppose  in  the  right-angled  tri- 
angle  ACE  the  sum  of  the  three  angles  is  equal  to  TT  —  a,  in  the  tri- 
angle AEF  equal  to  x  —  ft,  then  must  it  in  triangle  ACF  equal  TC  —  a 
—  ft,  where  a  and  ft  can  not  be  negative. 

Further,  let  the  angle  BAF  =  a,  AFC  =  b,  so  is  a  +/?  =  a  —  b;  now 
by  revolving  the  line  AF  away  from  the  perpendicular  AC  we  can  make 
the  angle  a  between  AF  and  the  parallel  AB  as  small  as  we  choose;  so 
also  can  we  lessen  the  angle  b,  consequently  the  two  angles  a  and  ft 
can  have  no  other  magnitude  than  a  =  0  and  ft  =  0. 

It  follows  that  in  all  rectilineal  triangles  the  sum  of  the  three  angles 
is  either  rr  and  at  the  same  time  also  the  parallel  angle  //  (p)  —  4  /T  for 
every  line  p,  or  for  all  triangles  this  sum  is  <  TT  and  at  the  same  time 
also  77(p)<  £  K. 

The  first  assumption  serves  as  foundation  for  the  ordinary  geometry  and 
plane  trigonometry. 

The  second  assumption  can  likewise  be  admitted  without  leading  to 
any  contradiction  in  the  results,  and  founds  a  new  geometric  science, 
to  which  I  have  given  the  name  Imaginary  Geometry,  and  which  I  in- 
tend here  to  expound  as  far  as  the  development  of  the  equations  be- 
tween the  sides  and  angles  of  the  rectilineal  and  spherical  triangle. 

23.  For  every  given  angle  a  there  is  a  line  p  such  that  fl  (p)  =  a. 
Let  AB  and  AC  (Fig.  10)  be  two  straight  lines  which  at  the  inter. 

section  point  A  make  the  acute  angle  a;  take  at  random  on  AB  a  point 


20 


THEORY    OF   PARALLELS. 


B';  from  this  point  drop  B'A'*at  right  angles  to  AC;  make  A' A*  r= 
AA';  erect  at  A'  the  perpendicular  A'B';  and  so  continue  until  a  per- 


FIG.  10. 


pendicular  CD  is  attained,  which  no  longer  intersects  AB.  This  must 
of  necessity  happen,  for  if  in  the  triangle  AA'B'  the  sum  of  all  three 
angles  is  equal  to  JT  —  a,  then  in  the  triangle  AB'  A*  it  equals  n  —  2a, 
in  triangle  AA'B"  less  than  TT  —  2a  (Theorem  20),  and  so  forth,  until 

it  finally  becomes  negative  and  thereby  shows  the  impossibility  of  con- 
structing the  triangle. 

The  perpendicular  CD  may  be  the  very  one  nearer  than  which  to  the 
point  A  all  others  cut  AB;  at  least  in  the  passing  over  from  those  that 
cut  to  those  not  cutting  such  a  perpendicular  FG  must  exist 

Draw  now  from  the  point  F  the  line  FH,  which  makes  with  FG  the 
acute  angle  HFG,  on  that  side  where  lies  the  point  A.  From  any  point 
H  of  the  line  FH  let  fall  upon  AC  the  perpendicular  HK,  whose  pro- 
longation consequently  must  cut  AB  somewhere  in  B,  and  so  makes  a 
triangle  A  KB,  into  which  the  prolongation  of  the  line  FH  enters,  and 
therefore  must  meet  the  hypothenuse  AB  somewhere  in  M.  Since  the 
angle  GFH  is  arbitrary  and  can  be  taken  as  small  as  we  wish,  therefore 
FG  is  parallel  to  AB  and  AF  =  p.  (Theorems  16  and  18.) 

One  easily  sees  that  with  the  lessening  of  p  the  angle  a,  increases,  while, 
for  p  =  0,  it  approaches  the  value  £TT;  with  the  growth  of  p  the  angle 
a  decreases,  while  it  continually  approaches  zero  for  p  =00  . 

Since  we  are  wholly  at  liberty  to  choose  what  angle  we  will  under- 


THEORY    OF    PARALLELS. 


21 


stand  by  the  symbol  77  (p)  when  the  line  p  is  expressed  by  a  negative 
number,  so  we  will  assume 


an  equation  which  shall  hold  for  all  values  of  p,  positive  as  well  as  neg- 
ative, and  for  p  =  0. 

24.      The  farther  parallel  lines  are  prolonged  on  the  side  of  their  paral- 
lelism, the  more  they  approach  one  another. 

If  to  the  line  AB  (Fig.  11)  two  perpendiculars  AC  =  BD  are  erected 
and  their  end-points  C  and  D  joined  by     „  D 

a  straight  line,  then  will  the  quadrilat- 
eral CABD  have  two  right  angles  at 
A  and  B,  but  two  acute  angles  at  C 
and  D  (Theorem  22)  which  are  equal 
to  one  another,  as  we  can  easily  see 


B. 


I 
by  thinking  the  quadrilateral  super-  FIQ.  11. 

imposed  upon  itself  so  that  the  line  BD  falls  upon  AC  and  AC  upon 
BD. 

Halve  AB  and  erect  at  the  mid-point  E  the  line  EF  perpendicular  to 
AB.  This  line  must  also  be  perpendicular  to  CE,  since  the  quadrilat- 
erals CAEF  and  FDBE  fit  one  another  if  we  so  place  one  on  the  other 
that  the  line  EF  remains  in  the  same  position.  Hence  the  line  CD  can 
not  be  parallel  to  AB,  but  the  parallel  to  AB  for  the  point  C,  namely 
CG,  must  incline  toward  AB  (Theorem  16)  and  cut  from  the  perpendic- 
ular BD  a  part  BG  <  CA. 

Since  C  is  a  random  point  in  the  line  CG,  it  follows  that  CG  itself 
nears  AB  the  more  the  farther  it  is  prolonged. 


22 


THEORY    OF    PARALLELS. 


25.      Two  straight  lines  which  are  parallel  to  a  third  are  also  parallel  to 
each  other. 


FIG.  12. 

"We  will  first  assume  that  the  three  lines  AB,  CD,  EF  (Fig.  12)  lie  in 
one  plane.  If  two  of  them  in  order,  AB  and  CD,  are  parallel  to  the 
outmost  one,  EF,  so  are  AB  and  CD  parallel  to  each  other.  In  order 
to  prove  this,  let  fall  from  any  point  A  of  the  outer  line  AB  upon  the 
other  outer  line  FE,  the  perpendicular  AE,  which  will  cut  the  middle 
line  CD  in  some  point  C  (Theorem  3),  at  an  angle  DCE  <  £  TT  on  the 
side  toward  EF,  the  parallel  to  CD  (Theorem  22). 

A  perpendicular  AG  let  fall  upon  CD  from  the  same  point,  A,  must 
fall  within  the  opening  of  the  acute  angle  ACG  (Theorem  9);  every 
other  line  AH  from  A  drawn  within  the  angle  BAG  must  cut  EF,  the 
parallel  to  AB,  somewhere  in  H,  how  small  soever  the  angle  BAH  may 
be;  consequently  will  CD  in  the  triangle  AEH  cut  the  line  AH  some- 
where in  K,  since  it  is  impossible  that  it  should  meet  EF.  If  AH  from 
the  point  A  went  out  within  the  angle  CAG,  then  must  it  cut  the  pro- 
longation of  CD  between  the  points  C  and  G  in  the  triangle  CAG. 
Hence  follows  that  AB  and  CD  are  parallel  (Theorems  16  and  18). 

Were  both  the  outer  lines  AB  and  EF  assumed  parallel  to  the  middle 
line  CD,  so  would  every  line  AK  from  the  point  A,  drawn  within  the 
angle  B AE,  cut  the  line  CD  somewhere  in  the  point  K,  how  small  soever 
the  angle  BAK  might  be. 

Upon  the  prolongation  of  AK  take  at  random  a  point  L  and  join  it 


THEORY   OF   PARALLELS.  23 

with  C  by  the  line  CL,  which  must  cut  EF  somewhere  in  M,  thus  mak- 
ing  a  triangle  MCE. 

The  prolongation  of  the  line  AL  within  the  triangle  MCE  can  cut 
neither  AC  nor  CM  a  second  time,  consequently  it  must  meet  EF  some- 
where  in  H;  therefore  AB  and  EF  are  mutually  parallel. 


FIG.  13. 

Now  let  the  parallels  AB  and  CD  (Fig.  13)  lie  in  two  planes  whose 
intersection  line  is  EF.  From  a  random  point  E  of  this  latter  let 
fall  a  perpendicular  EA  upon  one  of  the  two  parallels,  e.  g.,  upon  AB, 
then  from  A,  the  foot  of  the  perpendicular  EA,  let  fall  a  new  perpen- 
dicular AC  upon  the  other  parallel  CD  and  join  the  end-points  E  and  C 
of  the  two  perpendiculars  by  the  line  EC.  The  angle  BAG  must  be 
acute  (Theorem  22),  consequently  a  perpendicular  CG  from  C  let  fall 
upon  AB  meets  it  in  the  point  G  upon  that  side  of  CA  on  which  the 
lines  AB  and  CD  are  considered  as  parallel. 

Every  line  EH  [in  the  plane  FEAB],  however  little  it  diverges  from 
EF,  pertains  with  the  line  EC  to  a  plane  which  must  cut  the  plane  of 
the  two  parallels  AB  and  CD  along  some  line  CH.  This  latter  line  cuts 
AB  somewhere,  and  in  fact  in  the  very  point  H  which  is  common  to  all 
three  planes,  through  which  necessarily  also  the  line  EH  goes;  conse- 
quently EF  is  parallel  to  AB. 

In  the  same  way  we  may  show  the  parallelism  of  EF  and  CD. 

Therefore  the  hypothesis  that  a  line  EF  is  parallel  to  one  of  two  other 
parallels,  AB  and  CD,  is  the  same  as  considering  EF  as  the  intersection 
of  two  planes  in  which  two  parallels,  AB,  CD,  lie. 

Consequently  two  lines  are  parallel  to  one  another  if  they  are  parallel 
to  a  third  line,  though  the  three  be  not  co-planar. 

The  last  theorem  can  be  thus  expressed: 

Three  planes  intersect  in  lines  which  are  all  parallel  to  each  other  if  the 
parallelism  of  two  is  pre-supposed. 


24  THEORY    OF    PARALLELS. 

26.  Triangles  standing  opposite  to  one  another  on  the  sphere  are  equiva- 
lent in  surface. 

By  opposite  triangles  we  here  understand  such  as  are  made  on  both 
sides  of  the  center  by  the  intersections  of  the  sphere  with  planes;  in  such 
triangles,  therefore,  the  sides  and  angles  are  in  contrary  order. 

In  the  opposite  triangles  ABC  and  A' B'C'  (Fig.  14,  where  one  of 
them  must  be  looked  upon  as  represented  turned  about),  we  have  the 
•ides  AB  =  A'B',  BC  =  B'C',  CA  =  C'A',  and  the  corresponding  angles 


Fio.  14. 

at  the  points  A,  B,  C  are  likewise  equal  to  those  in  the  other  triangle  at 
th    points  A',  B',C'. 

Through  the  three  points  A,  B,  C,  suppose  a  plane  passed,  and  upon 
it  from  the  center  of  the  sphere  a  perpendicular  dropped  whose  pro- 
longations both  ways  cut  both  opposite  triangles  in  the  points  D  and  D' 
of  the  sphere.  The  distances  of  the  first  D  from  the  points  ABC,  in 
arcs  of  great  circles  on  the  sphere,  must  be  equal  (Theorem  12)  as  well 
to  each  other  as  also  to  the  distances  D'A',  D'B',  D'C',  on  the  other 
triangle  (Theorem  6),  consequently  the  isosceles  triangles  about  the  points 
D  and  D'  in  the  two  spherical  triangles  ABC  and  A'B'C'  are  congruent. 

In  order  to  judge  of  the  equivalence  of  any  two  surfaces  in  general, 
I  take  the  following  theorem  as  fundamental: 

Two  surfaces  are  equivalent  when  they  arise  from  the  mating  or  separating 
of  equal  parts. 

27.  A  three-sided  solid  angle  equals  the  half  sum  of  the  surface  angles 
'•ess  a  right-angle. 

In  the  spherical  triangle  ABC  (Fig.  15),  where  each  side  <  n,  desig- 
nate the  angles  by  A,  B,  C;  prolong  the  side  AB  so  that  a  whole  circle 
ABA'B'A  is  produced;  this  divides  the  sphere  into  two  equal  parts. 


THEORY    OF    PARALLELS. 


25 


In  that  half  in  which  is  the  triangle  ABC,  prolong  now  the  other  two 
sides  through  their  common  intersection  point  C  until  they  meet  the 
circle  in  A'  and  B'. 


In  this  way  the  hemisphere  is  divided  into  four  triangles,  ABC,  ACB', 
B'CA',  A'CB,  whose  size  may  be  designated  by  P,  X  Y,  Z.  It  is  evi 
dent  that  here  P  +  X  =  B,  P  -f  Z=  A. 

The  size  of  the  spherical  triangle  Y  equals  that  of  the  opposite  triangle 
ABC',  having  a  side  AB  in  common  with  the  triangle  P,  and  whose 
third  angle  C'  lies  at  the  end-point  of  the  diameter  of  the  sphere  which 
goes  from  C  through  the  center  D  of  the  sphere  (Theorem  26).  Hence 
it  follows  that 

P  -}-  Y  =  C,  and  since  P-f-X-f-Y-|-Z==7r,  therefore  we  have  also 


We  may  attain  to  the  same  conclusion  in  another  way,  based  solely 
upon  the  theorem  about  the  equivalence  of  surfaces  given  above.  (Theo- 
rem 26.) 

In  the  spherical  triangle  ABC  (Fig.  16),  halve  the  sides  AB  and  BC, 
and  through  the  mid-points  D  and 
B  draw  a  great  circle;  upon  this  let 
fall  from  A,  B,  C  the  perpendiculars 
AF,  BH,  and  CG.  If  the  perpendic- 
ular  from  B  falls  at  H  between  D  and 
E,  then  will  of  the  triangles  so  made 
BDH  =  AFD,  and  BHE  =  EGG  (The- 
orems  6  and  15),  whence  follows  that  Fio.  16. 

the  surface  of  the  triangle  ABC  equals  that  of  the  quadrilateral  AFGO 
(Theorem  26). 


26 


THEORY   OP   PARALLELS. 


If  the  point  H  coincides  with  the  middle  point  E  of  the  side  BC  (Fig. 
B  IV),  only  two  equal  right-angled  triangles,  ADF 
and  BDE,  are  made,  by  whose  interchange  the 
equivalence  of  the  surfaces  of  the  triangle  ABC 
and  the  quadrilateral  AFEC  is  established. 

If,  finally,  the  point  H  falls  outside  the  triangle 
ABC  (Fig.  18),  the  perpendicular  CG  goes,  in 
FIG.  17.  consequence,  through  the  triangle,  and  so  we  go 

over  from  the  triangle  ABC  to  the  quadrilateral  AFGC  by  adding  the 


triangle  FAD  =  DBH,  and  then  taking  away  the  triangle  CGE  =  EBEL 
Supposing  in  the  spherical  quadrilateral  AFGC  a  great  circle  passed 

through  the  points  A  and  G,  as  also  through  F  and  C,  then  will  their 

arcs  between  AG  and  FC  equal  one  another  (Theorem  15),  consequently 

also  the  triangles  FAC  and  ACG  be  congruent  (Theorem  15),  and  the 

angle  FAC  equal  the  angle  ACG. 

Hence  follows,  that  in  all  the  preceding  cases,  the  sum  of  all  three 

angles  of  the  spherical  triangle  equals  the  sum  of  the  two  equal  angles 

in  the  quadrilateral  which  are  not  the  right  angles. 

Therefore  we  can,  for  every  spherical  triangle,  in  which  the  sum  of 

the  three  angles  is  8,  find  a  quadrilateral  with  equivalent  surface,  in 

which  are  two  right  angles  and  two  equal  perpendicular  sides,  and 

where  the  two  other  angles  are  each  £S. 


THEORY    OF    PARALLELS. 


.27 


Let  now  ABCD  (Fig.  19)  be  the  spherical  quadrilateral,  where  the 
aides  AB  ~  DC  are  perpendicular  to  BC,  and  the  angles  A  and  D 
each  S. 


Fio.  19. 

Prolong  the  sides  AD  and  BC  until  they  cut  one  another  in  E,  and 
further  beyond  E,  make  DE  =  EF  and  let  fall  upon  the  prolongation 
of  BC  the  perpendicular  FG.  Bisect  the  whole  arc  BG  and  join  the 
mid -point  H  by  great-circle-arcs  with  A  and  F. 

The  triangles  EFG  and  DCE  are  congruent  (Theorem  15),  so  FG  = 
DC  =  AB. 

The  triangles  ABH  and  HGF  are  likewise  congruent,  since  they  are 
right  angled  and  have  equal  perpendicular  sides,  consequently  AH  and 
AF  pertain  to  one  circle,  the  arc  AHF  =  TT,  ADEF  likewise  =  n,  the 
angle  HAD  =  HFE  ==  £S  —  BAH  =  £S  —  HFG  =  |S  —  HFE— EFG 
—  |S— HAD-7r+£S;  consequently,  angle  HFE  =  £(S— ;r);  or  what 
is  the  same,  this  equals  the  size  of  the  lune  AHFDA,  which  again  is 
equal  to  the  quadrilateral  ABCD,  as  we  easily  see  if  we  pass  over  from 
the  one  to  the  other  by  first  adding  the  triangle  EFG  and  then  BAH 
and  thereupon  taking  away  the  triangles  equal  to  them  DCE  and  HFG. 

Therefore  £(S— TT)  is  the  size  of  the  quadrilateral  ABCD  and  at  the 
same  time  also  that  of  the  spherical  triangle  in  which  the  sum  of  the 
three  angles  is  equal  to  S. 


28 


THEORY    OF    PARALLELS. 


28.  If  three  planes  cut  each  other  in  parallel  lines,  then  the  sum  of  the 
three  surface  angles  equals  two  right  angles. 

Let  AA',  BB'  CC'  (Fig.  20)  be  three  parallels  made  by  the  inter- 
section  of  planes  (Theorem  25).  Take  upon  them  at  random  three 


FIG.  20. 

points  A,  B,  C,  and  suppose  through  these  a  plane  passed,  which  con- 
sequently will  cut  the  planes  of  the  parallels  along  the  straight  lines 
AB,  AC,  and  BC.  Further,  pass  through  the  line  AC  and  any  point 
D  on  the  BB',  another  plane,  whose  intersection  with  the  two  planes  of 
the  parallels  AA'  and  BB',  CC'  and  BB'  produces  the  two  lines  AD 
and  DC,  and  whose  inclination  to  the  third  plane  of  the  parallels  AA' 
and  CC'  we  will  designate  by  w. 

The  angles  between  the  three  planes  in  which  the  parallels  lie  will 
be  designated  by  X,  Y,  Z,  respectively  at  the  lines  A  A',  BB',  CC'; 
finally  call  the  linear  angles  BDC  =  a,  ADC  =  b,  ADB  =  c. 

About  A  as  center  suppose  a  sphere  described,  upon  which  the  inter- 
sections of  the  straight  lines  AC,  AD  A  A'  with  it  determine  a  spherical 
triangle,  with  the  sides  p,  q,  and  r.  Call  its  size  a.  Opposite  the  side 
q  lies  the  angle  w,  opposite  r  lies  X,  and  consequently  opposite  p  lies 
the  angle  ;r-|-2a — w— X,  (Theorem  27). 

In  like  manner  CA,  CD,  CC'  cut  a  sphere  about  the  center  C,  and 
determine  a  triangle  of  size  ft,  with  the  sides  p',  q',  r',  and  the  angles,  w 
opposite  q',  Z  opposite  r',  and  consequently  ;r-{-2/9— w—Z  opposite  p'. 

Finally  is  determined  by  the  intersection  of  a  sphere  about  D  with 
the  lines  DA,  DB,  DC,  a  spherical  triangle,  whose  sides  are  1,  m,  n,  and 
the  angles  opposite  them  w-\-Z—2ft,  w-\-"K — 2#,  and  Y.  Consequently 
its  size  <J=:£(X-fY-|-Z— n) — a— ft+w. 

Decreasing  w  lessens  also  the  size  of  the  triangles  a  and  ft,  so  that 
a-{-ft — w  can  be  made  smaller  than  any  given  number. 


THEORY    OF    PARALLELS. 


29 


In  the  triangle  o  can  likewise  the  sides  1  and  m  be  lessened  even  to 
vanishing  (Theorem  21),  consequently  the  triangle  3  can  be  placed  with 
one  of  its  sides  1  or  m  upon  a  great  circle  of  the  sphere  as  often  as  you 
choose  without  thereby  filling  up  the  half  of  the  sphere,  hence  d  van- 
ishes together  with  w;  whence  follows  that  necessarily  we  must  have 


29.  In  a  rectilineal  triangle,  the  perpendiculars  erected  at  the  mid-points 
of  the  sides  either  do  not  meet,  or  they  all  three  cut  each  other  in  one  point. 

Having  pre-supposed  in  the  triangle  ABC  (Fig.  21),  that  the  two  per- 
pendiculars ED  and  DF,  which  are  erected  upon  the  sides  AB  and  BC 
at  their  mid  points  E  and  F,  intersect  in  the  point  D,  then  draw  within 
the  angles  of  the  triangle  the  lines  DA,  DB,  DC. 

In  the  congruent  triangles  ADE  and  BDE  (Theorem  10),  we  have 
AD— ED,  thus  follows  also  that  BD=:CD;  the 
triangle  ADC  is  hence  isosceles,  consequently  the 
perpendicular  dropped  from  the  vertex  D  upon  the 
base  AC  falls  upon  G  the  mid-point  of  the  base. 

The  proof  remains  unchanged  also  in  the  case 
when  the  intersection  point  D  of  the  two  perpen- 
diculars ED  and  FD  falls  in  the  line  AC  itself,  or 
falls  without  the  triangle. 

In  case  we  therefore  pro-suppose  that  two  of  those  perpendiculars  do 
not  intersect,  then  also  the  third  can  not  meet  with  them. 

30.  The  perpendiculars  which  are  erected  upon  the  sides  of  a  rectilineal 
triangle  at  their  mid-points,  must  all  three  be  parallal  to  each  other,  so  soon 
as  the  parallelism  of  two  of  them  is  pre-supposed. 

In  the  triangle  ABC  (Fig.  22)  let  the  lines  DE,  FG,  HK,  be  erected 
perpendicular  upon  the  sides  at  their  mid- 
points D,  F,  H.  "We  will  in  the  first  place 
assume  that  the  two  perpendiculars  DE  and 
FG  are  parallel,  cutting  the  line  AB  in  L 
and  M,  and  that  the  perpendicular  HK  lies 
between  them.  Within  the  angle  BLE  draw 
from  the  point  L,  at  random,  a  straight  line 
LG,  which  must  cut  FG  somewhere  in  G,  Fio.  22. 

how  small  soever  the  angle  of  deviation  GLE  may  be.     (Theorem  16). 


30  THEORY    OF    PARALLELS. 

Since  in  the  triangle  LGM  the  perpendicular  HK  <!an  not  meet  with 
MG  (Theorem  29),  therefore  it  must  cut  LG  somewhere  in  P,  whence 
follows,  that  HK  is  parallel  to  DE  (Theorem  16),  and  to  MG  (Theorems 
18  and  25). 

Put  the  side  BC  =  2a,  AC  =  2b,  AB  =  2c,  and  designate  the  an- 
gles  opposite  these  sides  by  A,  B,  C,  then  we  have  in  the  case  just 
considered 

A=/7(b)-/7(c), 
B=/7(a)_/7(c), 


as  one  may  easily  show  with  help  of  the  lines  AA',  BB',  CC',  which 
are  drawn  from  the  points  A,  B,  C,  parallel  to  the  perpendicular  HK 
and  consequently  to  both  the  other  perpendiculars  DE  and  FG  (Theo- 
rems 23  and  25). 

Let  now  the  two  perpendiculars  HK  and  FG  be  parallel,  then  can 
the  third  DE  not  cut  them  (Theorem  29),  hence  is  it  either  parallel  to 
them,  or  it  cuts  AA'. 

The  last  assumption  is  not  other  than  that  the  angle 
C>/7(a)+/7(b.) 

If  we  lessen  this  angle,  so  that  it  becomes  equal  to  77  (a)  +//(b), 
while  we  in  that  way  give  the  line  AC  the  new  position  CQ,  (Fig.  23), 
and  designate  the  size  of  the  third  side  BQ  by  2c',  then  must  the  angle 
CBQ  at  the  point  B,  which  is  increased,  in  accordance  with  what  is 
proved  above,  be  equal  to 

/7(a)—  /7(c')>/7(a)—  //(c), 
whence  follows  c'  >c  (Theorem  23). 

A 


FIG.  23. 

In  the  triangle  ACQ  are,  however,  the  angles  at  A  and  Q  equal, 
hence  in  the  triangle  ABQ  must  the  angle  at  Q  be  greater  than  that  at 
the  point  A,  consequently  is  AB>BQ,  (Theorem  9);  that  is  c>c'. 

31.  We  call  boundary  line  (oricycle)  that  curve  lying  in  a  plane  for 
which  all  perpendiculars  erected  at  the  mid-points  of  chords  are  parallel  to 
each  other. 


THEORY    OF    PARALLELS.  31 

In  conformity  with  this  definition  we  can  represent  the  generation  of 
a  boundary  line,  if  we  draw  to  a  given  line  AB  (Fig.  24)  from  a  given 

H! 


FIG.  24. 

point  A  in  it,  making  different  angles  CAB=  /7(a),  chords  AC  =  2a; 
the  end  C  of  such  a  chord  will  lie  on  the  boundary  line,  whose  points 
we  can  thus  gradually  determine. 

The  perpendicular  DE  erected  upon  the  chord  AC  at  its  mid-point  D 
will  be  parallel  to  the  line  AB,  which  we  will  call  the  Axis  of  the  bound- 
ary line.  In  like  manner  will  also  each  perpendicular  FG  erected  at  the 
mid-point  of  any  chord  AH,  be  parallel  to  AB,  consequently  must  this 
peculiarity  also  pertain  to  every  perpendicular  KL  in  general  which  is 
erected  at  the  mid-point  K  of  any  chord  CH,  between  whatever  points 
C  and  H  of  the  boundary  line  this  may  be  drawn  (Theorem  30).  Such 
perpendiculars  must  therefore  likewise,  without  distinction  from  AB, 
be  called  Axes  of  the  boundary  line. 

32.  A  circle  with  continually  increasing  radius  merges  into  the  boundary 
line. 

Given  AB  (Fig.  25)  a  chord  of  the  boundary  line;  draw  from  the 
end-points  A  and  B  of  the  chord  two  axes 
AC   and    BF,    which    consequently  will 
make   with  the  chord  two  equal  angles 
BAG  =  ABF  =  a  (Theorem  31). 

Upon  one  of  these  axes  AC,  take  any- 
where the  point  E  as  center  of  a  circle, 

and  draw  the  arc  AF  from  the  initial  point      A  K  —  w 

A  of  the  axis  AC  to  its  intersection  point  FIG.  25. 

F  with  the  other  axis  BF. 

The  radius  of  the  circle,  FE,  corresponding  to  the  point  F  will  make 
on  the  one  side  with  the  chord  AF  an  angle  AFE=^9,  and  on  the 


32  THEORY    OF    PARALLELS. 

other  side  with  the  axis  BF,  the  angle  EFD  =  f.  It  follows  that  the 
angle  between  the  two  chords  BAF  =  a — fi<fij^y — a  (Theorem  22); 
whence  follows,  a — /9<i7" 

Since  now  however  the  angle  f  approaches  the  limit  zero,  as  well  in 
consequence  of  a  moving  of  the  center  E  in  the  direction  AC,  when  F 
remains  unchanged,  (Theorem  21),  as  also  in  consequence  of  an  ap- 
proach of  F  to  B  on  the  axis  BF,  when  the  center  E  remains  in  its 
position  (Theorem  22),  so  it  follows,  that  with  such  a  lessening  of  the 
angle  7-,  also  the  angle  a — ft,  or  the  mutual  inclination  of  the  two  chords 
AB  and  AF,  and  hence  also  the  distance  of  the  point  B  on  the  bound- 
ary line  from  the  point  F  on  the  circle,  tends  to  vanish. 

Consequently  one  may  also  call  the  boundary -line  a  circle  with  in- 
finitely great  radius. 

33.     Let  A  A'  —  BB'  =  x  (Figure  26).  be  two  lines  parallel  toward 
the  side  from  A  to  A',  which  parallels  serve 
as  axes  for  the  two  boundary  arcs  (arcs  on 
two  boundary  lines)  AB=s,  A'B'  =*',  then  is 

s'  —  Be  —  * 

where  e  is  independent  of  the  arcs  s,  s'  and  of  Fio.  26. 

the  straight  hue  x,  the  distance  of  the  arc  s'  from  s. 

In  order  to  prove  this,  assume  that  the  ratio  of  the  arc  s  to  t'  is 
equal  to  the  ratio  of  the  two  whole  numbers  n  and  TO. 

Between  the  two  axes  AA',  BB'  draw  yet  a  third  axis  CC',  which 
so  cuts  off  from  the  arc  AB  a  part  AC  =  t  and  from  the  arc  A'B'  on 
the  same  side,  a  part  A'C'  =  t'.  Assume  the  ratio  of  t  to  s  equal  to 
that  of  the  whole  numbers  p  and  q,  so  that' 

n  p 

s=  —  s',     t=  —  s. 
m  q 

Divide  now  *  by  axes  into  nq  equal  parts,  then  will  there  be  mq  such 
parts  on  sf  and  np  on  t. 

However  there  correspond  to  these  equal  parts  on  s  and  t  likewise 
equal  parts  on  s'  and  t1,  consequently  we  have 

t'       s' 

t         s 

Hence  also  wherever  the  two  arcs  t  and  t'  may  be  taken  between  the 
two  axes  AA;  and  BB;,  the  ratio  of  t  to  t'  remains  always  the  same,  as 


THEORY    OF    PARALLELS. 


long  as  the  distance  x  between  them  remains  the  same.     If  we  there- 
fore for  x=i,  put  s—  es',  then  we  must  have  for  every  x 

s'  =  se~x. 

Since  6  is  an  unknown  number  only  subjected  to  the  condition  e>i, 
and  further  the  linear  unit  for  x  may  be  taken  at  will,  therefore  we  may, 
for  the  simplification  of  reckoning,  so  choose  it  that  by  e  is  to  be  un- 
derstood the  base  of  Napierian  logarithms. 

"We  may  here  remark,  that  s'=  0  for  «=  GO  ,  hence  not  only  does 
the  distance  between  two  parallels  decrease  (Theorem  24),  but  with  the 
prolongation  of  the  parallels  toward  the  side  of  the  parallelism  this  at 
last  wholly  vanishes.  Parallel  lines  have  therefore  the  character  of 
asymptotes. 

34.  Boundary  surface  (prisphere)  we  call  that  surface  which  arises 
from  the  revolution  of  the  boundary  line  about  one  of  its  axes,  which, 
together  with  all  other  axes  of  the  boundary-line,  will  be  also  an  axis 
of  the  boundary-surface. 

A  chord  is  inclined  at  equal  angles  to  such  axes  drawn  through  its  end- 
points,  wheresoever  these  two  end-points  may  be  taken  on  the  boundary-surface. 

Let  A,  B,  C,  (Fig.  27),  be  three  points  on  the  boundary-surface; 


FIG.  27. 

A  A',  the  axis  of  revolution,  BB'  and  CC'  two  other  axes,  hence  AB 
and  AC  chords  to  which  the  axes  are  inclined  at  equal  angles  A'AB 
=B'BA,  A'AC  =  C'CA  (Theorem  31.) 


34  THEORY    OF    PARALLELS. 

Two  axes  BB',  CO',  drawn  through  the  end-points  of  the  third  chord 
BC,  are  likewise  parallel  and  lie  in  one  plane,  (Theorem  25). 

A  perpendicular  DD'  erected  at  the  mid -point  D  of  the  chord  AB 
and  in  the  plane  of  the  two  parallels  A  A',  BB',  must  be  parallel  to  the 
three  axes  AA',  BB',  CO',  (Theorems  23  and  25);  just  such  a  perpen- 
dicular BE'  upon  the  chord  AC  in  the  plane  of  the  parallels  A  A',  CC' 
will  be  parallel  to  the  three  axes  AA',  BB',  CC',  and  the  perpendicular 
DD'.  Let  now  the  angle  between  the  plane  in  which  the  parallels  AA' 
and  BB'  lie,  and  the  plane  of  the  triangle  ABC  be  designated  by  77  (a), 
where  a  may  be  positive,  negative  or  null.  If  a  is  positive,  then  erect 
FD  =  a  within  the  triangle  ABC,  and  in  its  plane,  perpendicular  upon 
the  chord  A  B  at  its  mid-point  D. 

"Were  a  a  negative  number,  then  must  FD  =  a  be  drawn  outside  the 
triangle  on  the  other  side  of  the  chord  AB;  when  a— 0,  the  point  F 
coincides  with  D. 

In  all  cases  arise  two  congruent  right-angled  triangles  AFD  and  DFB, 
consequently  we  have  FA  =  FB. 

Erect  now  at  F  the  line  FF'  perpendicular  to  the  plane  of  the  tri- 
angle ABC. 

Since  the  angle  D'DF  =  /7(a),  and  DF=a,  so  FF'  is  parallel  to 
DD'  and  the  line  EE',  with  which  also  it  lies  in  one  plane  perpendicu- 
lar to  the  plane  of  the  triangle  ABC. 

Suppose  now  in  the  plane  of  the  parallels  EE',  FF'  upon  EF  the  per- 
pendicular  EK  erected,  then  will  this  be  also  at  right  angles  to  the  plane 
of  the  triangle  ABC,  (Theorem  13),  and  to  the  line  AE  lying  in  this 
plane,  (Theorem  11);  and  consequently  must  AE,  which  is  perpendicu- 
lar to  EK  and  EE',  be  also  at  the  same  time  perpendicular  to  FE, 
(Theorem  1 1).  The  triangles  AEF  and  FEC  are  congruent,  since  they 
are  right-angled  and  have  the  sides  about  the  right  angles  equal,  hence  is 

AF  =  FC  =  FB. 

A  perpendicular  from  the  vertex  F  of  the  isosceles  triangle  BFC  let 
fall  upon  the  base  BC,  goes  through  its  mid-point  G;  a  plane  passed 
through  this  perpendicular  FG  and  the  line  FF'  must  be  perpendicular 
to  the  plane  of  the  triangle  ABC,  and  cuts  the  plane  of  the  parallels 
BB',  CC',  along  the  line  GG;,  which  is  likewise  parallel  to  BB'  and 
CC',  (Theorem  25);  since  now  CG  is  at  right  angles  to  FG,  and  hence 
at  the  same  time  also  to  GG',  so  consequently  is  the  angle  C'CG 
=  B'BG,  (Theorem  23). 


THEORY    OF   PARALLELS. 


35 


Hence  follows,  that  for  the  boundary-surface  each  of  the  axes  may 
be  considered  as  axis  of  revolution. 

Principal-plane  we  will  call  each  plane  passed  through  an  axis  of  the 
boundary  surface. 

Accordingly  every  Principal-plane  cuts  the  boundary-surface  in  the 
boundary  line,  while  for  another  position  of  the  cutting  plane  this  in. 
tersection  is  a  circle. 

Three  principal  planes  which  mutually  cut  each  other,  make  with 
each  other  angles  whose  sum  is  TT,  (Theorem  28). 

These  angles  we  will  consider  as  angles  in  the  boundary-triangle 
whose  sides  are  arcs  of  the  boundary-line,  which  are  made  on  the  bound- 
ary surface  by  the  intersections  with  the  three  principal  planes.  Con- 
sequently the  same  interdependence  of  the  angles  and  sides  pertains  to 
the  boundary-triangles,  that  is  proved  in  the  ordinary  geometry  for  the 
rectilineal  triangle. 

35.  In  what  follows,  we  will  designate  the  size  of  a  line  by  a  letter 
with  an  accent  added,  e.  g.  a/,  in  order  to  indicate  that  this  has  a  rela. 
tion  to  that  of  another  line,  which  is  represented  by  the  same  letter 
without  accent  x,  which  relation  is  given  by  the  equation 


Let  now  ABC  (Fig.  28)  be  a  rectilineal  right-angled  triangle,  where 
the  hypothenuse  AB  =  c,  the  other  sides  AC  =  b,  BC  =  a,  and  the 


FIG.  28. 


angles  opposite  them  are 


36  THEORY    OF    PARALLELS. 

At  the  point  A  erect  the  line  AA'  at  right  angles  to  the  plane  of  the 
triangle  ABC,  and  from  the  points  B  and  C  draw  BB'  and  CO'  parallel 
toAA'. 

The  planes  in  which  these  three  parallels  lie  make  with  each  other 
the  angles:  /7(a)  at  AA',  a  right  angle  at  CC'  (Theorems  11  and  13), 
consequently  //(«')  at  BB'  (Theorem  28). 

The  intersections  of  the  lines  BA,  BC,  BB'  with  a  sphere  described 
about  the  point  B  as  center,  determine  a  spherical  triangle  mnk,  in  which 
the  sides  are  mn  =  77(c),  Ten  =  /I  (ft),  mJc  =  /7(a)  and  the  opposite  angle? 
are  /7(b),  //(«'),  $x. 

Therefore  we  must,  with  the  existence  of  a  rectilineal  triangle  whoa.. 
sides  are  a,  b,  c  and  the  opposite  angles  U  (a),  IJ(ft)  fa  also  admit  th-s 
existence  of  a  spherical  triangle  (Fig.  29)  with  the  sides  77  (c), 
/7(a)  and  the  opposite  angles  77(b),  //(«')>  fa 


FIG.  29. 

Of  these  two  triangles,  however,  also  inversely  the  existence  of  the 
spherical  triangle  necessitates  anew  that  of  a  rectilineal,  which  in  con- 
sequence,  also  can  have  the  sides  a,  a',  ft,  and  the  oppsite  angles  /7(b'), 
/7(c),  fa 

Hence  we  may  pass  over  from  a,  b,  c,  a,  ft,  to  b,  a,  c,  ft,  a,  and  also  to  a, 
a',  ft,  b',  c. 

Suppose  through  the  point  A  (Fig.  28)  with  AA'  as  axis,  a  bound- 
ary-surface  passed,  which  cuts  the  two  other  axes  BB',  CC',  in  B*  and 
C",  and  whose  intersections  with  the  planes  the  parallels  form  a  bound- 
ary-triangle, whose  sides  are  B"C'=p,  C'A  — y,  B'A— r?  and  the 
angles  opposite  them  //(«),  //(«'),  fa  and  where  consequently  (Theo- 
rem 34): 

p  =  r  sin  IJ(a),  q  =  r  cos  77(«). 

Now  break  the  connection  of  the  three  principal-planes  along  the  line 
BB' ,  and  turn  them  out  from  each  other  so  that  they  with  all  the  lines 
lying  in  them  come  to  lie  in  one  plane,  where  consequently  the  arcs  p, 
g,  r  will  unite  to  a  single  arc  of  a  boundary-line,  which  goes  through  the 


THEORY    OF    PARALLELS.  37 

point  A  and  has  A  A'  for  axis,  in  such  a  manner  that  (Fig.  '30)  on  the 
one  side  will  lie,  the  arcs  q  and  p,  the  side  b  of  the  triangle,  which  is 


Fro.  30. 

perpendicular  to  AA'  at  A,  the  axis  CC'  going  from  the  end  of  b  par- 
allel to  A  A'  and  through  C'  the  union  point  of  p  and  q,  the  side  a  per- 
pendicular to  CC'  at  the  point  C,  and  from  the  end-point  of  a  the  axis 
BB'  parallel  to  A  A'  which  goes  through  the  end-point  B'  of  the  arc  p. 

On  the  other  side  of  AA'  will  lie,  the  side  c  perpendicular  to  AA'  at 
the  point  A,  and  the  axis  BB'  parallel  to  AA',  and  going  through  the 
end-point  B*  of  the  arc  r  remote  from  the  end  point  of  b. 

The  size  of  the  line  CC"  depends  upon  b,  which  dependence  we  will 
express  by  CC'  =/(b). 

In  like  manner  we  will  have  BB'  =/(c). 

If  we  describe,  taking  CC '  as  axis,  a  new  boundary  line  from  the 
point  C  to  its  intersection  D  with  the  axis  BB'  and  designate  the  arc 
CD  by  t,  then  is  ED  =  f  (a). 

BB'=:  BD-f-DB'  =  BD-fCC',  consequently 


Moreover,  we  perceive,  that  (Theorem  33) 


If  the  perpendicular  to  the  plane  of  the  triangle  ABC  (Fig.  28)  were 
erected  at  B  instead  of  at  the  point  A,  then  would  the  lines  c  and  r  remain 
the  same,  the  arcs  q  and  t  would  change  to  t  and  q,  the  straight  lines  a 


So  THEORY    OP   PARALLELS. 

and  b  into  b  and  a,  and  the  angle  77  (a)  into  77(^9),  consequently  we 
would  have 


q=  rsn 
whence  follows  by  substituting  the  value  of  q, 

cos  77  (a)  =  sin  77  (/9)  e*»>, 
and  if  we  change  a  and  ft  into  b'  and  c, 

sin77(b)^sin77(c)e«a>; 
further,  by  multiplication  with  e-^b) 

sin  II  (b)  e/<b>=  sin  77  (c)  e*e> 
Hence  follows  also 

sin  II  (a)  e-K»>—  sin  77  (b)  e^). 

Since  now,  however,  the  straight  lines  a  and  b  are  independent  of 
one  another,  and  moreover,  for  b=0,  /(b)=0,  77(b)=r^r,  so  we  have 
for  every  straight  line  a 

e-/<»>=sin77(a). 
Therefore, 

sin  77  (c)  =  sin  77  (a)  sin  77  (b), 
sin  77  (/9)  =  cos  77  (a)  sin  77  (a). 
Hence  we  obtain  besides  by  mutation  of  the  letters 
sin  77  (a)  =  cos  77  Q3)  sin  77  (b), 
cos  77  (b)  =  cos  77  (c)  cos  77  (a), 
cos  II  (a)  =  cos  77(c)  cos  77(£). 

If  we  designate  in  the  right-angled  spherical  triangle  (Fig.  29)  the 
sides  77(c),  77  (/3),  77  (a),  with  the  opposite  angles  77  (b),  77  (<//),  ^7  t^e 
letters  a,  b,  c,  A,  B,  then  the  obtained  equations  take  on  the  form  of 
those  which  we  know  as  proved  in  spherical  trigonometry  for  the  right- 
angled  triangle,  namely, 

sin  a=sin  c  sin  A, 
sin  b=sin  c  sin  B, 
cos  A=cos  a  sin  B, 
cos  B=cos  b,  sin  A, 
cos  c=cos  a,  cos  b; 

from  which  equations  we  can  pass  over  to  those  for  all  spherical  tri* 
angles  in  general 

Hence  spherical  trigonometry  is  not  dependent  upon  whether  in  a 


THEORY    OF    PARALLELS. 


39 


rectilineal  triangle  the  sum  of  the  three  angles  is  equal  to  two  right 
angles  or  not 

36.  We  will  now  consider  anew  the  right-angled  rectilineal  triangle 
ABC  (Fig.  31),  in  which  the  sides  are  a,  b,  c,  and  the  opposite  angles 
B(a\  77(0),  \it. 

Prolong  the  hypothenuse  c  through 
the  point  B,  and  make  BD=y9;  at  the 
point  D  erect  upon  BD  the  perpendicu- 
lar DD;,  which   consequently  will  be 
parallel  to  BB' ,  the  prolongation  of  the 
side  a  beyond  the  point  B.     Parallel  to 
DIX  from  the  point  A  draw  AA',  which 
is  at  the  same  time  also  parallel  to  CB', 
(Theorem  25),  therefore  is  the  angle 
A'AD=77(c+£), 
A' AC = 77  (b),  consequently 
77(b)=77(a)+77(c+$. 


FIG.  32. 


Fio.  31. 

If  from  B  we  lay  off  /9  on  the  hypoth- 
enuse  c,  then  at  the  end  point  D,  (Fig. 
32),  within  the  triangle  erect  upon  AB 
the  perpendicular  DD',  and  from  the 
point  A  parallel  to  DIX  draw  AA',  so 
will  BC  with  its  prolongation  CC'  be  the 
third  parallel;  then  is,  angle  CAA'=/7 
(b),  DAA'=77(c — £),  consequently 77(c — 
/?)=77(a)-f77(b).  The  last  equation  is 
then  also  still  valid,  when  c=ft,  or  c<^. 

If  c=/?  (Fig.  33),  then  the  perpendicu- 
ular  A  A'  erected  upon  AB  at  the  point  A 


40  THEORY   OF   PARALLELS. 

is  parallel  to  the  side  BC^a,  with  its  prolongation,  CC',  consequently 


3. 

we  have  77  (a)  +  77  (b)  =£7:,  whilst  also  77(c—  ft)=£7T,  (Theorem  23). 

If  c<ft,  then  the  end  of  ft  falls  beyond  the  point  A  at  D  (Fig.  34) 
upon  the  prolongation  of  the  hypothenuse  AB.  Here  the  perpendicu- 
lar DD'  erected  upon  AD,  and  the  line  AA'  parallel  to  it  from  A,  will 

i*       likewise  be  parallel  to  the  side  BC=a, 
with  its  prolongation  CC'. 

Here  we  have  the  angle  DAA'  —  JJ 
(ft  —  c)»  consequently 


(Theorem  23). 

The  combination  of  the  two  equations 

found  gives, 

277(b)=77(c-ft)+77(c+ft), 
277(a)=77(e—  ft)—  77(c+ft), 

whence  follows 

cos  77(b)  _cos  [  £77(c—  j9)-H-  /7c- 

cos  //(.*)  ""cos  [  i/7(c—  p)—±  / 

Substituting  here  the  value,  (Theo- 


/     rem  35) 


Fio.  34. 


cos  II  (b) 


=cos/7(c), 


cos  77  (a) 
we  have  [tan  £77 (c)]2=tan  £77 (c— ft)  tan  £77 (c-f  ft). 

Since  here  ft  is  an  arbitrary  number,  as  the  angle  77  (ft)  at  the  one 


THEORY    OF    PARALLELS. 


41 


side  of  c  may  be  chosen  at  will  between  the  limits  0  and  ^TT,  conse- 
quently ft  between  the  limits  0  and  oo  ,  so  we  may  deduce  by  taking 
consecutively  ft=c,  2c,  3c,  &c.,  that  for  every  positive  number  n,  [tan£ 


If  we  consider  n  as  the  ratio  of  two  lines  x  and  c,  and  assume  that 


then  we  find  for  every  line  x  in  general,  whether  it  be  positive  or  nega- 
tive, tan|/7(x)=e—  * 

where  e  may  be  any  arbitrary  number,  which  is  greater  than  unity, 
since  77(x)=0  for  x—  <»  . 

Since  the  unit  by  which  the  lines  are  measured  is  arbitrary,  so  we 
may  also  understand  by  e  the  base  of  the  Napierian  Logarithms. 

37.     Of  the  equations  found  above  in  Theorem  35  it  is  sufficient  to 
know  the  two  following, 

sin  /7(c)=.sin  77  (a)  sin  77  (b) 
sin/7  (a)=sin  /7(b)  cos  /7(£), 

applying  the  latter  to  both  the  sides  a  and  b  about  the  right  angle,  in 
order  irom  the  combination  to  deduce  the  remaining  two  of  Theorem 
35,  without  ambiguity  of  the  algebraic  sign,  since  here  all  angles  are 
acute. 

In  a  similar  manner  we  attain  the  two  equations 
(1.)    tan  /7(c)=sin  /7(a)  tan  /7(a), 
(2.)    cos  /7(a)=cos  /7(c)  cos 


We  will  now  consider  a  rectilineal  triangle  whose  sides  are  a,  b,  c, 
(Fig.  35)  and  the  opposite  angles  A,  B,  C. 

If  A  and  B  are  acute  angles,  then  the 
perpendicular  p  from  the  vertex  of  the 
angle  C  falls  within  the  triangle  and  cuts 
the  side  c  into  two  parts,  x  on  the  side  of 
the  angle  A  and  c — x  on  the  side  of  the 
Fro.  35.  'angle  B.  Thus  arise  two  right-angled 

triangles,  for  which  we  obtain,  by  application  of  equation  (1), 
tan  /7(a)=sin  B  tan  77(p), 
tan  /7(b)=sin  A  tan  77  (p), 


42 


THEORY    OP   PARALLELS. 


which  equations  remain  unchanged  also  when  one  of  the  angles,  «.  y.  B, 
is  a  right  angle  (Fig.  36)  or  and  obtuse  angle  (Fig.  37). 


C  H 

FIG.  37. 


Fio.  36. 
Therefore  we  have  universally  for  every  triangle 

(3.)    sin  A  tan  /7(a)=sin  B  tan  /7(b). 

For  a  triangle  with  acute  angles  A,  B,  (Fig.  35)  we  have  also  (Equa- 
tion 2), 

cos  /7(x)=cos  A  cos  77(b), 
cos  /7(c — x)=cos  B  cos  77(a) 

which  equations  also  relate  to  triangles,  in  which  one  of  the  angles  A 
or  B  is  a  right  angle  or  an  obtuse  angle. 

As  example,  for  B=£;r  (Fig.  36)  we  must  take  x=c,  the  first  equa- 
tion then  goes  over  into  that  which  we  have  found  above  as  Equation  2, 
the  other,  however,  is  self-sufficing. 

For  B>£?r  (Fig.  37)  the  first  equation  remains  unchanged,  instead 
of  the  second,  however,  we  must  write  correspondingly 
cos  77(x— c)=cos  (TT— B)  cos  77(a); 
but  we  have  cos  /7(x — c) = — cos  77(c — x) 

(Theorem  23),  and  also  cos  (x— B)== — cos  B. 

If  A  is  a  right  or  an  obtuse  angle,  then  must  c — x  and  x  be  put  for 
x  and  c — x,  in  order  to  carry  back  this  case  upon  the  preceding. 

In  order  to  eliminate  x  from  both  equations,  we  notice  that  (Theo- 
rem 36) 

l-[tanj/7(c-x)]» 
~l+[tan|/7(c— x)]? 


cos/7(c 


1— [tan  |/7(c)]8[cot  j/7( 
:  l+[tan£/7(c)]*[co 
COB  /7(c) — cos/7(x) 
1— cos  /7(c)cos/7(x) 


THEORY    OP    PARALLELS.  43 

If  we  substitute  here  the  expression  for  cos  77(x),  cos/7(c  —  *),  we  ob- 
tain 

j,.  ,  _  COB  77(a)  cos  B-j-cos77(b)  cos  A 
«"AC>=  i^coe//^)  coe/7(b)  cosA  cosB 
whence  follows 


cos  /7(a)  cosB^  <**  %)~<**A  cos77(b) 

1  —  cosA  cos/7(b)  cos  77(c) 
and  finally 

[sin/7(c)]»=[l  —  cosBcos/7(c)cos/7(a)][l    -cosAcos77(b)cos/7(c)] 
In  the  same  way  we  must  also  have 

w 

[sin  77(a)  ]»  =[1—  -cos  C  cos  77  (a)  cos  77  (b)  ]  [1—  cos  B  cos  77  (c)  cos  77(a)  ] 
[sin  77(b)  ]8  =[1—  -cos  A  cos  /7  (b)  cos  /7(c)  ]  [1—  cos  C  coe  /7(a)  cos  77(b)] 
From  these  three  equations  we  find 

[sin77(b)]«[sin77(c)]*  .,  ., 

[siny/(a)]«       -=[!—  cosAcos/7(b)cos/7(c)]«. 

Hence  follows  without  ambiguity  of  sign, 

IT  /i.x        rr,  -^   .  sin/7(b)sin/7Yc) 

(5.)  cos  A  cos  //  (b)  cos  //(c)  H  --  .v  '      .  —  ±±=  1. 

sin  /7  (a) 

If  we  substitute  here  the  value  of  sin  77  (c)  corresponding  to  equa- 
tion (3.) 


sin  JJ(c)=^-—  tan  H  (a)  cos  JJ  (c) 


then  we  obtain 


cos  Hfc)  =.  cos /7(a)  sinC 


sin  A  sin  /7  (b)-j-cos  A  sin  C  cos  77  (a)  cos  77  (b); 
but  by  substituting  this  expression  for  cos  77  (c)  in  equation  (4), 

(6.)  cot  A  sin  C  sin  77  (b)+cos  0= 


cos77(a) 
By  elimination  of  sin  77(b)  with  help  of  the  equation  (3)  comes 

cos  77  (a)  cos  A  . 

—  ^cosC^l  --  r-R-sinCsin77(a). 
cos  77  (b)  sin  B 

In  the  meantime  the  equation  (6)  gives  by  changing  the  letters, 


cos 


77(a) 


cos  77  (b) 


==cotB  sin  C  sin  77(a)-J-cosC. 


44  THEORY    OF    PARALLELS. 

From  the  last  two  equations  follows, 

.  _        _     sin  B  sin  C 

(7.)  cos  A4-C08  B  cos  C— — : — „.  . 

sm77(a) 

All  four  equations  for  the  interdependence  of  the  sides  a,  b,  c,  and 
the  opposite  angles  A,  B,  C,  in  the  rectilineal  triangle  will  therefore  be, 
[Equations  (3),  (5),  (6),  (7).] 

sin  A  tan  77  (a)  =  sin  B  tan  77  (b), 

sin  77  (b)  sin  77  (c) 
cos  A  cos  77  (b)  cos  77  (c)  -\ 


cot  A  sin  C  sin  77  (b)  -f-  cos  C  = 
cos  A  -f-  cos  B  cos  C  = 


77(a) 
cos  77  (b) 
cos  77  (a)  ' 
sin  B  sin  C 


sin  77  (a) 

If  the  sides  a,  b,  c,  of  the  triangle  are  very  small,  we  may  content  our* 
selves  with  the  approximate  determinations.     (Theorem  36.) 
cot  77  (a)  =  a, 
sin  77  (a)  =  1  —  |a« 
cos  77  (a)  =  a, 
and  in  like  manner  also  for  the  other  sides  b  and  c. 

The  equations  8  pass  over  for  such  triangles  into  the  following: 
b  sin  A  =  a  sin  B, 
a8  =b*  +  c8  —  2bc  cos  A, 
a  sin  (  A  -j-  C)  =  b  sin  A, 
cos  A  -f  cos  (B  +  C)  =  0. 

Of  these  equations  the  first  two  are  assumed  in  the  ordinary  geom- 
etry; the  last  two  lead,  with  the  help  of  the  first,  to  the  conclusion 


Therefore  the  imaginary  geometry  passes  over  into  the  ordinary,  when 
we  suppose  that  the  sides  of  a  rectilineal  triangle  are  very  small. 

I  have,  in  the  scientific  bulletins  of  the  University  of  Kasan,  pub- 
lished  certain  researches  in  regard  to  the  measurement  of  curved  lines, 
of  plane  figures,  of  the  surfaces  and  the  volumes  of  solids,  as  well  as  in 
relation  to  the  application  of  imaginary  geometry  to  analysis. 

The  equations  (8)  attain  for  themselves  already  a  sufficient  foundation 
for  considering  the  assumption  of  imaginary  geometry  as  possible. 
Hence  there  is  no  means,  other  than  astronomical  observations,  to  use 


THEORY   OP   PARALLELS.  46 

for  judging  of  the  exactitude  which  pertains  to  the  calculations  of  the 
ordinary  geometry. 

This  exactitude  is  very  far-reaching,  as  I  have  shown  in  one  of  my 
Investigations,  so  that,  for  example,  in  triangles  whose  sides  are  attain' 
able  for  our  measurement,  the  sum  of  the  three  angles  is  not  indeed  dif- 
ferent from  two  right-angles  by  the  hundredth  part  of  a  second. 

In  addition,  it  is  worthy  of  notice  that  the  four  equations  (8)  of 
plane  geometry  pass  over  into  the  equations  for  spherical  triangles,  if 
we  put  a  ^/—  1,  b  <y/—  1,  c  ^/—  1,  instead  of  the  sides  a,  b,  c;  with  this 
change,  however,  we  must  also  put 


gn 


cos  (a), 
(V- 


tan  n  (a)  =-.  -     . 

sina(-Y/—  1), 

and  similarly  also  for  the  sides  b  and  c. 

In  this  manner  we  pass  over  from  equations  (8)  to  the  following: 
sin  A  sin  b  =  sin  B  sin  a, 
cosa  =  cosb  cosc-j-sinb  sine  cos  A, 
cot  A  sin  C  -{-  cos  C  cos  b  =  sin  b  cot  a, 
cos  A  =  cos  a  sin  B  sin  C  —  cos  B  cos  C. 


TRANSLATOR'S  APPENDIX. 


ELLIPTIC   GEOMETRY. 

Gauss  himself  never  published  aught  upon  this  fascinating  subject, 
Geometry  Non-Euclidean;  but  when  the  most  extraordinary  pupil  of 
his  long  teaching  life  came  to  read  his  inaugural  dissertation  before  the 
Philosophical  Faculty  of  the  University  of  Goettingen,  from  the  three 
themes  submitted  it  was  the  choice  of  Gauss  which  fixed  upon  the  one 
"Ueber  die  Hypothesen  welche  der  Geometric  zu  Grunde  liegen." 

Gauss  was  then  recognized  as  the  most  powerful  mathematician  in  the 
world.  I  wonder  if  he  saw  that  here  his  pupil  was  already  beyond  him, 
when  in  his  sixth  sentence  Riemann  says,  "  therefore  space  is  only  a 
special  case  of  a  three-fold  extensive  magnitude,"  and  continues: 
"From  this,  however,  it  follows  of  necessity,  that  the  propositions  of 
geometry  can  not  be  deduced  from  general  magnitude-ideas,  but  that 
those  peculiarities  through  which  space  distinguishes  itself  from  other 
thinkable  threefold  extended  magnitudes  can  only  be  gotten  from  ex- 
perience. Hence  arises  the  problem,  to  find  the  simplest  facts  from 
which  the  metrical  relations  of  space  are  determmable  —  a  problem 
which  from  the  nature  of  the  thing  is  not  fully  determinate;  for  there 
may  be  obtained  several  systems  of  simple  facts  which  suffice  to  deter- 
mine the  metrics  of  space;  that  of  Euclid  as  weightiest  is  for  the  pres- 
ent aim  made  fundamental.  These  facts  are,  as  all  facts,  not  necessary, 
but  only  of  empirical  certainty;  they  are  hypotheses.  Therefore  one 
can  investigate  their  probability,  which,  within  the  limits  of  observation, 
of  course  is  very  great,  and  after  this  judge  of  the  allowability  of  their 
extension  beyond  the  bounds  of  observation,  as  well  on  the  side  of  the 
immeasurably  great  as  on  the  side  of  the  immeasurably  small." 

Riemann  extends  the  idea  of  curvature  to  spaces  of  three  and  more 
dimensions.  The  curvature  of  the  sphere  is  constant  and  positive,  and 
on  it  figures  can  freely  move  without  deformation.  The  curvature  of 
the  plane  is  constant  and  zero,  and  on  it  figures  slide  without  stretching. 
The  curvature  of  the  two-dimentlonal  space  of  Lobachevski  and 

[47] 


48  THEORY    OF    PARALLELS. 

Bolyai  completes  the  group,  being  constant  and  negative,  and  in  it  fig- 
ures can  move  without  stretching  or  squeezing.  As  thus  corresponding 
to  the  sphere  it  is  called  the  pseudo-sphere. 

In  the  space  in  which  we  live,  we  suppose  we  can  move  without  de- 
formation. It  would  then,  according  to  Riemann,  be  a  special  case  of 
a  space  of  constant  curvature.  We  presume  its  curvature  nulL  At 
once  the  supposed  fact  that  our  space  does  not  interfere  to  squeeze  us 
or  stretch  us  when  we  move,  is  envisaged  as  a  peculiar  property  of  our 
space.  But  is  it  not  absurd  to  speak  of  space  as  interfering  with  any- 
thing? If  you  think  so,  take  a  knife  and  a  raw  potato,  and  try  to  cut 
it  into  a  seven-edged  solid. 

Further  on  In  this  astonishing  discourse  comes  the  epoch-making  idea, 
that  though  space  be  unbounded,  it  is  not  therefore  infinitely  great. 
Riemann  says:  "In  the  extension  of  space-constructions  to  the  im- 
measurably great,  the  unbounded  is  to  be  distinguished  from  the  in- 
finite; the  first  pertains  to  the  relations  of  extension,  the  latter  to  the 
size-relations. 

"That  our  space  is  an  unbounded  three-fold  extensive  manifoldness,  is 
a  hypothesis,  which  is  applied  in  each  apprehension  of  the  outer  world, 
according  to  which,  in  each  moment,  the  domain  of  actual  perception  is 
filled  out,  and  the  possible  places  of  a  sought  object  constructed,  and 
which  in  these  applications  is  continually  confirmed.  The  unbounded- 
ness  of  space  possesses  therefore  a  greater  empirical  certainty  than  any 
outer  experience.  From  this  however  the  Infinity  in  no  way  follows. 
Rather  would  space,  if  one  presumes  bodies  independent  of  place,  that 
is  ascribes  to  it  a  constant  curvature,  necessarily  be  finite  so  soon  as  this 
curvature  had  ever  so  small  a  positive  value.  One  would,  by  extend- 
ing the  beginnings  of  the  geodesies  lying  in  a  surface-element,  obtain 
an  unbounded  surface  with  constant  positive  curvature,  therefore  a  sur- 
face which  in  a  homaloidal  three-fold  extensive  manifoldness  would 
take  the  form  of  a  sphere,  and  so  is  finite." 

Here  we  have  for  the  first  time  in  human  thought  the  marvelous  per- 
ception that  universal  space  may  yet  be  only  finite. 

Assume  that  a  straight  line  is  uniquely  determined  by  two  points,  but 
take  the  contradictory  of  the  axiom  that  a  straight  line  is  of  infinite 
size;  then  the  straight  line  returns  into  itself,  and  two  having  inter- 
sected get  back  to  that  intersection  point. 


BIBLIOGRAPHY. 


A  bibliography  of  non-Euclidean  literature  down  to  the  year  1878 
was  given  by  Halsted,  "American  Journal  of  Mathematics,"  vols.  i,  ii, 
containing  81  authors  and  174  titles,  and  reprinted  in  the  collected 
works  of  Lobachevski  (Kazan,  1886)  giving  124  authors  and  272  titles. 
This  was  incorporated  in  Bonola's  Bibliography  of  the  Foundations  of 
Geometry  (1899)  reprinted  (1902)  at  Kolozsvar  in  the  Bolyai  Memorial 
Volume.  In  1911  appeared  the  volume:  Bibliography  of  Non-Euclidean 
Geometry  by  Duncan  M.  Y.  Sommerville;  London,  Harrison  and  Sons. 

The  Introduction  says :  "The  present  work  was  begun  about  nine 
years  ago.  It  was  intended  as  a  continuation  of  Halsted's  bibliography, 
but  it  soon  became  evident  that  the  growth  of  the  subject  rendered  such 
diffuse  treatment  practically  impossible,  and  short  abstracts  of  the 
works  would  have  to  be  dispensed  with.  The  object  is  to  produce  as 
far  as  possible  a  complete  repository  of  the  titles  of  all  works  from 
the  earliest  times  up  to  the  present  which  deal  with  the  extended 
conception  of  space,  and  to  form  a  guide  to  the  literature  in  an  easily 
accessible  form.  It  includes  the  theory  of  parallels,  non-euclidean  ge- 
ometry, the  foundations  of  geometry,  and  space  of  n  dimensions." 

In  1913  Teubner  issued  in  two  parts  Paul  Stackers  important  book: 
Wolfgang  und  Johaun  Bolyai.  Geometrische  Untersuchungen.  John  com- 
pares Lobachevski's  researches  with  his  own.  The  profound  philosophic 
import  of  non-euclidean  geometry  forms  an  integrant  part  of  "The 
Foundations  of  Science,"  by  H.  Poincar6;  Vol.  I  of  the  series  Science 
and  Education,  The  Science  Press,  New  York  City,  1914.  The  Transac- 
tions of  the  Royal  Society  of  Canada,  Vol.  XII,  Section  III,  contains 
a  striking  Presidential  Address  by  Alfred  Baker  on  The  Foundations 
of  Geometry.  Of  the  cognate  works  issued  by  The  Open  Court  Pub. 
Co.,  we  mention  only  Euclid's  Parallel  Postulate  by  Withers.  Scores 
of  errors  are  pointed  out  in  "Non-Euclidean  Geometry  in  the  Encyclo- 
paedia Britannica,"  Science,  May  10,  1912. 

And  now  at  last  the  theory  of  relativity  has  made  non-euclidean 
geometry  a  powerful  machine  for  advance  in  physics. 

Says  Vladimir  Varicak  in  a  remarkable  lecture,  "Ueber  die  nicht- 

[49] 


50  BIBLIOGRAPHY 

euklidische   Interpretation  der  Relatlvtheorie,"    (Jahresber.   D.  Math. 
Ver.,  21,  103-127), 

I  postulated  that  the  phenomena  happened  in  a  Lobachevski  space, 
and  reached  by  very  simple  geometric  deduction  the  formulas  of  the 
relativity  theory-  Assuming  non-euclidean  terminology,  the  formulas 
of  the  relativity  theory  become  not  only  essentially  simplified,  but 
capable  of  a  geometric  interpretation  wholly  analogous  to  the  inter- 
pretation of  the  classic  theory  in  the  euclidean  geometry.  And  this 
analogy  often  goes  so  far,  that  the  very  wording  of  the  theorems  of 
the  classic  theory  may  be  left  unchanged. 


QA.  LobacnevsKii  -Geometrical  researches  on  the 
68 ^  theory  of  parallels 

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